High-low frequency slaving and regularity issues in the 3D Navier-Stokes equations

The old idea that an infinite dimensional dynamical system may have its high modes or frequencies slaved to low modes or frequencies is re-visited in the context of the $3D$ Navier-Stokes equations. A set of dimensionless frequencies $\{\tilde{\Omega}_{m}(t)\}$ are used which are based on $L^{2m}$-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that the $\tilde{\Omega}_{m}$ ($m>1$) are slaved to $\tilde{\Omega}_{1}$ (the global enstrophy) in the form $\tilde{\Omega}_{m} = \tilde{\Omega}_{1}\mathcal{F}_{m}(\tilde{\Omega}_{1})$. This is shaped by the constraint of two H\"older inequalities and a time average from which emerges a form for $\mathcal{F}_{m}$ which has been observed in previous numerical Navier-Stokes and MHD simulations. When written as a phase plane in a scaled form, this relation is parametrized by a set of functions $1 \leq \lambda_{m}(\tau) \leq 4$, where curves of constant $\lambda_{m}$ form the boundaries between tongue-shaped regions. In regions where $2.5 \leq \lambda_{m} \leq 4$ and $1 \leq \lambda_{m} \leq 2$ the Navier-Stokes equations are shown to be regular\,: numerical simulations appear to lie in the latter region. Only in the central region $2 < \lambda_{m} < 2.5$ has no proof of regularity been found.Comment: 10 pages; 2 figure

Integrability of the hyperbolic reduced Maxwell-Bloch equations for strongly correlated Bose-Einstein condensates

We derive and study the hyperbolic reduced Maxwell-Bloch equations (HRMB) which acts as a simplified model for the dynamics of strongly correlated Bose-Einstein condensates. A proof of their integrability is found by the derivation of a Lax pair which is valid for both the hyperbolic and standard cases of the reduced Maxwell-Bloch equations. The origin of the latter lies in quantum optics. We derive explicit solutions of the HRMB equations that correspond to kinks propagating on the Bose-Einstein condensate (BEC). These solutions are different from Gross-Pitaevskii solitons because the nonlinearity of the HRMB equations arises from the interaction of the BEC and excited atoms

Intermittency in solutions of the three-dimensional Navier-Stokes equations

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Nonlinear effects in buoyancy-driven variable-density turbulence

We consider the time dependence of a hierarchy of scaled L²ᵐ-norms D_m,ω and D_m,θ of the vorticity ω =∇ x u and the density gradient ∇θ, where θ = log.(ρ*/ ρ*₀), in a buoyancy-driven turbulent flow as simulated by Livescu & Ristorcelli (J. Fluid Mech., vol. 591, 2007, pp. 43–71). Here, ρ* (x,t) is the composition density of a mixture of two incompressible miscible fluids with fluid densities ρ*₂ > ρ*₁, and ρ*₀ is a reference normalization density. Using data from the publicly available Johns Hopkins turbulence database, we present evidence that the L²-spatial average of the density gradient can reach extremely large values at intermediate times, even in flows with low Atwood number At = (ρ*₂ - ρ*₁)/(ρ*₂ + ρ*₁) = 0.05, implying that very strong mixing of the density field at small scales can arise in buoyancy-driven turbulence. This large growth raises the possibility that the density gradient ∇θ might blow up in a finite time.We acknowledge, with thanks, the staff of IPAM UCLA where this collaboration began in the Autumn of 2014 on the programme ‘Mathematics of Turbulence’. We would also like to thank C. Doering and D. Livescu for useful discussions. All of the numerical data used are from the JHTDB (Livescu et al. 2014), a publicly available DNS database. For more information, please see http://turbulence.pha.jhu.edu/. We also thank the referees for suggesting substantial improvements

Variable density model for the Rayleigh-Taylor instability and its transformation to the diffusive, inhomogeneous, incompressible Navier-Stokes equations

It is shown how the variable density model that governs the Rayleigh-Taylor instability for the miscible mixing of two incompressible fluids can be transformed into a diffusive version of the inhomogeneous, incompressible Navier-Stokes equations forced by gradients of the composition density ρ of the mixing layer. This demonstrates how buoyancy-driven flows drive and enhance Navier-Stokes turbulence. The role of the potential vorticity q = ω ⋅ ∇ ρ is also discussed

The cultural capitalists: notes on the ongoing reconfiguration of trafficking culture in Asia

Most analysis of the international flows of the illicit art market has described a global situation in which a postcolonial legacy of acquisition and collection exploits cultural heritage by pulling it westwards towards major international trade nodes in the USA and Europe. As the locus of consumptive global economic power shifts, however, these traditional flows are pulled in other directions: notably for the present commentary, towards and within Asia

Low temperature growth and optical properties of alpha-Ga2O3 deposited on sapphire by plasma enhanced atomic layer deposition

Plasma enhanced atomic layer deposition was used to deposit thin films of Ga2O3 on to c-plane sapphire substrates using triethylgallium and O2 plasma. The influence of substrate temperature and plasma processing parameters on the resultant crystallinity and optical properties of the Ga2O3 films were investigated. The deposition temperature was found to have a significant effect on the film crystallinity. At temperatures below 200°C amorphous Ga2O3 films were deposited. Between 250°C and 350°C the films became predominantly α-Ga2O3. Above 350°C the deposited films showed a mixture of α-Ga2O3 and ε-Ga2O3 phases. Plasma power and O2 flow rate were observed to have less influence over the resultant phases present in the films. However, both parameters could be tuned to alter the strain of the film. Ultraviolet transmittance measurements on the Ga2O3 films showed that the bandgaps ranges from 5.0 eV to 5.2 eV with the largest bandgap of 5.2 eV occurring for the α-Ga2O3 phase deposited at 250°C

Phylogeny of Prokaryotes and Chloroplasts Revealed by a Simple Composition Approach on All Protein Sequences from Complete Genomes Without Sequence Alignment

The complete genomes of living organisms have provided much information on their phylogenetic relationships. Similarly, the complete genomes of chloroplasts have helped to resolve the evolution of this organelle in photosynthetic eukaryotes. In this paper we propose an alternative method of phylogenetic analysis using compositional statistics for all protein sequences from complete genomes. This new method is conceptually simpler than and computationally as fast as the one proposed by Qi et al. (2004b) and Chu et al. (2004). The same data sets used in Qi et al. (2004b) and Chu et al. (2004) are analyzed using the new method. Our distance-based phylogenic tree of the 109 prokaryotes and eukaryotes agrees with the biologists tree of life based on 16S rRNA comparison in a predominant majority of basic branching and most lower taxa. Our phylogenetic analysis also shows that the chloroplast genomes are separated to two major clades corresponding to chlorophytes s.l. and rhodophytes s.l. The interrelationships among the chloroplasts are largely in agreement with the current understanding on chloroplast evolution