Miguel D. Bustamante

3D Euler about a 2D symmetry plane

Initial results from new calculations of interacting anti-parallel Euler vortices are presented. The objective is to understand the origins of singular scaling presented by Kerr (1993) with different core profiles, develop more robust analysis for identifying singular behaviour and to develop criteria for when calculations should be terminated. If a localised three-dimensional perturbation is added to an analytic vortex core, then smoothed with a symmetric hyperviscous filter (the first two steps of the initialisation of Kerr (1993)), then an anomalous region of negative vorticity forms in the lee of the primary vortex as in Fig. 1. This is the primary difference between between the initial condition of Kerr (1993) and Hou and Li (2006). At late times the anomalous vorticity becomes sandwiched between the two primary vortices, requiring extra resolution. The additional Chebyshev mapping of Kerr (1993) removes this anomaly, which we have reproduced with a purely Fourier expansion by adding positive vorticity as a function of z as in Fig 2. Results from a similar initial condition with additional initial stretching of the shape of the vortex cores in x are given. The tools for identifying singular versus regular behaviour have been reexamined as well as the criteria for terminating the calculations. Kerr (1993) assumed the simplest power law (or similar) growth of peak vorticity, enstrophy and enstrophy production. A new approach is to: a) Infer a singular time tc for a quantity and its directly determined time derivative, b) Use this to estimate the power-law behaviour as t → tc and c) Find the pre-factor. If the estimated singular time tc increases as the calculation proceeds, then regular behaviour is favoured. Fig. 3 shows results using the global enstrophy Ω and enstrophy production Ωp = Ω from the data used for Kerr (1993). A new conclusion si that Ω ∼ (tc − t)−γΩ as t→ tc with γΩ ≈ 0.5 rather than the logarithmic growth suggested by Kerr (1993). New results are consistent, but do not go as far in time. It is found that numerical depletion of circulation in the symmetry plane is the best indication of when the calculations become underresolved. Current results imply that such simple behaviour either does not occur or occurs at such late times that it would be unreachable using Fourier methods. Figure 1: ωy in the symmetry plane from a test initial condition with only step 1) and 2) (a high-wavenumber filter). Note the large negative vorticity in the lee (right) of the primary vortex as in HouLi (2006) (Fig. 2) and how this is entrained underneath the primary vortex.

The three-dimensional Euler equations : singular or non-singular?

One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought.

Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries

We investigate the ideal and incompressible magnetohydrodynamic (MHD) equations in three space dimensions for the development of potentially singular structures. The methodology consists in implementing the fourfold symmetries of the Taylor-Green vortex generalized to MHD, leading to substantial computer time and memory savings at a given resolution; we also use a regridding method that allows for lower-resolution runs at early times, with no loss of spectral accuracy. One magnetic configuration is examined at an equivalent resolution of 6144(3) points and three different configurations on grids of 4096(3) points. At the highest resolution, two different current and vorticity sheet systems are found to collide, producing two successive accelerations in the development of small scales. At the latest time, a convergence of magnetic field lines to the location of maximum current is probably leading locally to a strong bending and directional variability of such lines. A novel analytical method, based on sharp analysis inequalities, is used to assess the validity of the finite-time singularity scenario. This method allows one to rule out spurious singularities by evaluating the rate at which the logarithmic decrement of the analyticity-strip method goes to zero. The result is that the finite-time singularity scenario cannot be ruled out, and the singularity time could be somewhere between t=2.33 and t=2.70. More robust conclusions will require higher resolution runs and grid-point interpolation measurements of maximum current and vorticity.

Dynamics of nonlinear resonances in Hamiltonian systems

It is well known that the dynamics of a Hamiltonian system depends crucially on whether or not it possesses nonlinear resonances. In the generic case, the set of nonlinear resonances consists of independent clusters of resonantly interacting modes, described by a few low-dimensional dynamical systems. We formulate and prove a new theorem on integrability which allows us to show that most frequently met clusters are described by integrable dynamical systems. We argue that construction of clusters can be used as the base for the Clipping method, substantially more effective for these systems than the Galerkin method. The results can be used directly for systems with cubic Hamiltonian.

Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem.

Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to 4096^{3}. The results are analyzed in terms of the classical analyticity-strip method and Beale, Kato, and Majda (BKM) theorem. A well-resolved acceleration of the time decay of the width of the analyticity strip δ(t) is observed at the highest resolution for 3.7<t<3.85 while preliminary three-dimensional visualizations show the collision of vortex sheets. The BKM criterion on the power-law growth of the supremum of the vorticity, applied on the same time interval, is not inconsistent with the occurrence of a singularity around t≃4. These findings lead us to investigate how fast the analyticity-strip width needs to decrease to zero in order to sustain a finite-time singularity consistent with the BKM theorem. A simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the BKM theorem with the analyticity-strip method. It is shown that a finite-time blowup can exist only if δ(t) vanishes sufficiently fast at the singularity time. In particular, if a power law is assumed for δ(t) then its exponent must be greater than some critical value, thus providing a new test that is applied to our 4096^{3} Taylor-Green numerical simulation. Our main conclusion is that the numerical results are not inconsistent with a singularity but that higher-resolution studies are needed to extend the time interval on which a well-resolved power-law behavior of δ(t) takes place and check whether the new regime is genuine and not simply a crossover to a faster exponential decay.

Apoptosis is directly related to intracellular amyloid accumulation in a cell line derived from the cerebral cortex of a trisomy 16 mouse, an animal model of Down syndrome

Human Down syndrome (DS) represents the most frequent cause of mental retardation associated to a genetic condition. DS also exhibits a characteristic early onset of neuropathology indistinguishable from that observed in Alzheimers disease (AD), namely the deposition of the beta-amyloid peptide. Early endosomal dysfunction has been described in individuals with DS and AD, suggesting an important role of this subcellular compartment in the onset and progression of the pathology. On the other hand, cholesterol activates the amyloidogenic processing pathway for the amyloid precursor protein, and the lipoprotein receptor-related peptide interacts with the beta-amyloid peptide. In the present work, using cell lines derived from the cortex of both normal and trisomy 16 mice (Ts16), an animal model of DS, we showed that the application of exogenous beta-amyloid has cytotoxic effects, expressed in decreased viability and increased apoptosis. Supplementation of the culture media with cholesterol associated to lipoprotein increased cell viability in both cell lines, but apoptosis decreased only in the normal cell line. Further, intracellular beta-amyloid content was elevated in trisomic cells following cholesterol treatment, with higher values in the trisomic cell line. Immunocytochemical detection showed intracellular accumulation of exogenous beta-amyloid in Rab4-positive compartments, which are known to be associated to endosomal recycling. The results suggest that the intracellular beta-amyloid pool plays a central role in apoptosis-mediated cell death in the trisomic condition.

Externally forced triads of resonantly interacting waves : boundedness and integrability properties

Abstract We revisit the problem of a triad of resonantly interacting nonlinear waves driven by an external force applied to the unstable mode of the triad. The equations are Hamiltonian, and can be reduced to a dynamical system for 5 real variables with 2 conservation laws. If the Hamiltonian, H, is zero we reduce this dynamical system to the motion of a particle in a one-dimensional time-independent potential and prove that the system is integrable. Explicit solutions are obtained for some particular initial conditions. When explicit solution is not possible we present a novel numerical/analytical method for approximating the dynamics. Furthermore we show analytically that when H = 0 the motion is generically bounded. That is to say the waves in the forced triad are bounded in amplitude for all times for any initial condition with the single exception of one special choice of initial condition for which the forcing is in phase with the nonlinear oscillation of the triad. This means that the energy in the forced triad generically remains finite for all time despite the fact that there is no dissipation in the system. We provide a detailed characterisation of the dependence of the period and maximum energy of the system on the conserved quantities and forcing intensity. When H ≠ 0 we reduce the problem to the motion of a particle in a one-dimensional time-periodic potential. Poincare sections of this system provide strong evidence that the motion remains bounded when H ≠ 0 and is typically quasi-periodic although periodic orbits can certainly be found. Throughout our analyses, the phases of the modes in the triad play a crucial role in understanding the dynamics.

Generalized Eulerian-Lagrangian description of Navier-Stokes dynamics

Generalized equations of motion for the Weber-Clebsch potentials that reproduce Navier-Stokes dynamics are derived. These depend on a new parameter, with the dimension of time, and reduce to the Ohkitani and Constantin equations in the singular special case where the new parameter vanishes. Let us recall that Ohkitani and Constantin found that the diffusive Lagrangian map became noninvertible under time evolution and required resetting for its calculation. They proposed that high frequency of resetting was a diagnostic for vortex reconnection. Direct numerical simulations are performed. The Navier-Stokes dynamics is well reproduced at small enough Reynolds number without resetting. Computation at higher Reynolds numbers is achieved by performing resettings. The interval between successive resettings is found to abruptly increase when the new parameter is varied from 0 to a value much smaller than the resetting interval.

Precession and recession of the rock'n'roller

We study the dynamics of a spherical rigid body that rocks and rolls on a plane under the effect of gravity. The distribution of mass is non-uniform and the centreofmassdoesnotcoincidewiththegeometriccentre. Thesymmetriccase, withmomentsofinertia I1 = I2 <I 3,isintegrableandthemotioniscompletely regular. Threeknownconservationlawsarethetotalenergy E,Jellett’squantity QJ and Routh’s quantity QR. When the inertial symmetry I1 = I2 is broken, even slightly, the character of the solutions is profoundly changed and new types of motion become possible. We derive the equations governing the general motion and present analytical and numerical evidence of the recession, or reversal of precession, that has been observed in physical experiments. We present an analysis of recession in terms of critical lines dividing the (QR ,Q J ) plane into four dynamically disjoint zones. We prove that recession implies the lack of conservation of Jellett’s and Routh’s quantities, by identifying individualreversalsascrossingsoftheorbit (QR(t), QJ (t))throughthecritical lines. Consequently, a method is found to produce a large number of initial conditions so that the system will exhibit recession.

Quadratic invariants for discrete clusters of weakly interacting waves

We consider discrete clusters of quasi-resonant triads arising from a Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a functionally independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular M × N matrix A with entries 1, −1 and 0. In particular, we prove that the number of independent quadratic invariants is equaltoJ ≡ N −M ∗ N −M,whereM ∗ isthenumberoflinearlyindependent rows in A. Thus, the problem of finding all independent quadratic invariants is reduced to a linear algebra problem in the Hamiltonian case. We establish that the properties of the quadratic invariants (e.g., locality) are related to the topological properties of the clusters (e.g., types of linkage). To do so, we formulate an algorithm for decomposing large clusters into smaller ones and show how various invariants are related to certain parts of a cluster, including thebasicstructuresleadingtoM ∗ < M.Weillustrateourfindings bypresenting examples from the Charney‐Hasegawa‐Mima wave model, and by showing a classification of small (up to three-triad) clusters.