Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations

In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% \rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi} 0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.Comment: 5 more figures can be found in the corresponding journal paper (J. Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm Equations, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundar

The tachocline revisited

The solar tachocline is a shear layer located at the base of the solar convection zone. The horizontal shear in the tachocline is likely turbulent, and it is often assumed that this turbulence would be strongly anisotropic as a result of the local stratification. What role this turbulence plays in the tachocline dynamics, however, remains to be determined. In particular, it is not clear whether it would result in a turbulent eddy diffusivity, or anti-diffusivity, or something else entirely. In this paper, we present the first direct numerical simulations of turbulence in horizontal shear flows at low Prandtl number, in an idealized model that ignores rotation and magnetic fields. We find that several regimes exist, depending on the relative importance of the stratification, viscosity and thermal diffusivity. Our results suggest that the tachocline is in the stratified turbulence regime, which has very specific properties controlled by a balance between buoyancy, inertia, and thermal diffusion.Comment: Invited review for the meeting Dynamics of the Sun and Stars: Honoring the Life and Work of Michael J. Thompson (Boulder, Colorado, 24-26 September 2019

Solar rotation rate and its gradients during cycle 23

Available helioseismic data now span almost the entire solar activity cycle 23 making it possible to study solar-cycle related changes of the solar rotation rate in detail. In this paper we study how the solar rotation rate, in particular, the zonal flows change with time. In addition to the zonal flows that show a well known pattern in the solar convection zone, we also study changes in the radial and latitudinal gradients of the rotation rate, particularly in the shear layer that is present in the immediate sub-surface layers of the Sun. In the case of the zonal-flow pattern, we find that the band indicating fast rotating region close to the equator seems to have bifurcated around 2005. Our investigation of the rotation-rate gradients show that the relative variation in the rotation-rate gradients is about 20% or more of their average values, which is much larger than the relative variation in the rotation rate itself. These results can be used to test predictions of various solar dynamo models.Comment: To appear in ApJ. Fig 5 has been corrected in this versio

Tools to support the self assessment of the performance of Food Safety Management Systems

Changes in food supply chains, health and demographic situations, lifestyle and social situations, environmental conditions, and increased legislative requirements have led to significant efforts in the development of quality and safety management systems in agribusiness and food industry worldwide (Ropkins and Beck, 2000; Efstratiadis, Karirti, and Arvanitoyannis, 2000; Jacxsens, et al, 2009a, Luning and Marcelis, 2009a). Nowadays, companies have implemented various quality assurance (QA) guidelines and standards, such as GMP and HACCP guidelines (like General Principles of food hygiene (Codex Alimentarius 2003), GFSI guidance document (GFSI (2007), and quality assurance standards (like ISO 9001:2008 (2008), ISO22000:2005 (2005), BRC (2008), and IFS (2007) into their company own food safety management system. The performance of such systems in practice is, however, still variable. Moreover, the continuous pressure on food safety management system (FSMS) performance and the dynamic environment wherein the systems operate (such as emerging pathogens, changing consumer demands, developments in preservation techniques) require that they can be systematically analysed to determine opportunities for improvement (Wallace, et al, 2005; Manning et al, 2006; Van der Spiegel et al, 2006; Cornier et al, 2007; Luning et al, 2009a). Within the European project entitled ‘PathogenCombat- EU FOOD-CT-2005-007081’ various tools have been developed to support food companies and establishments in systematically analysing and judging their food safety management system and its microbiological performance as basis for strategic choices on interventions to improve the FSMS performance. This chapter describes briefly principles of the major tools that have been developed and some others, which are still under still under construction

Deeply penetrating banded zonal flows in the solar convection zone

Helioseismic observations have detected small temporal variations of the rotation rate below the solar surface corresponding to the so-called `torsional oscillations' known from Doppler measurements of the surface. These appear as bands of slower and faster than average rotation moving equatorward. Here we establish, using complementary helioseismic observations over four years from the GONG network and from the MDI instrument on board SOHO, that the banded flows are not merely a near-surface phenomenon: rather they extend downward at least 60 Mm (some 8% of the total solar radius) and thus are evident over a significant fraction of the nearly 200 Mm depth of the solar convection zone.Comment: 4 pages, 4 figures To be published in ApJ Letters (accepted 3/3/2000

Continuous unidirectional fiber reinforced composites: Fabrication and testing

The study of the anisotropic mechanical properties of an inexpensively fabricated composite with continuous unidirectional fibers and a clear matrix was investigated. A method has been developed to fabricate these composites with aluminum fibers and a polymer matrix. These composites clearly demonstrate the properties of unidirectional composites and cost less than five dollars each to fabricate

Communication in cross-cultural consultations in primary care in Europe: the case for improvement. The rationale for the RESTORE FP 7 project

The purpose of this paper is to substantiate the importance of research about barriers and levers to the implementation of supports for cross-cultural communication in primary care settings in Europe. After an overview of migrant health issues, with the focus on communication in cross-cultural consultations in primary care and the importance of language barriers, we highlight the fact that there are serious problems in routine practice that persist over time and across different European settings. Language and cultural barriers hamper communication in consultations between doctors and migrants, with a range of negative effects including poorer compliance and a greater propensity to access emergency services. It is well established that there is a need for skilled interpreters and for professionals who are culturally competent to address this problem. A range of professional guidelines and training initiatives exist that support the communication in cross-cultural consultations in primary care. However, these are commonly not implemented in daily practice. It is as yet unknown why professionals do not accept or implement these guidelines and interventions, or under what circumstances they would do so. A new study involving six European countries, RESTORE (REsearch into implementation STrategies to support patients of different ORigins and language background in a variety of European primary care settings), aims to address these gaps in knowledge. It uses a unique combination of a contemporary social theory, normalisation process theory (NPT) and participatory learning and action (PLA) research. This should enhance understanding of the levers and barriers to implementation, as well as providing stakeholders, with the opportunity to generate creative solutions to problems experienced with the implementation of such interventions

Exact, E=0, Solutions for General Power-Law Potentials. I. Classical Orbits

For zero energy, $E=0$, we derive exact, classical solutions for {\em all} power-law potentials, $V(r)=-\gamma/r^\nu$, with $\gamma>0$ and $-\infty <\nu<\infty$. When the angular momentum is non-zero, these solutions lead to the orbits $\r(t)= [\cos \mu (\th(t)-\th_0(t))]^{1/\mu}$, for all $\mu \equiv \nu/2-1 \ne 0$. When $\nu>2$, the orbits are bound and go through the origin. This leads to discrete discontinuities in the functional dependence of $\th(t)$ and $\th_0(t)$, as functions of $t$, as the orbits pass through the origin. We describe a procedure to connect different analytic solutions for successive orbits at the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. Also, we explain why they all must violate the virial theorem. The unbound orbits are also discussed in detail. This includes the unusual orbits which have finite travel times to infinity and also the special $\nu = 2$ case.Comment: LaTeX, 27 pages with 12 figures available from the authors or can be generated from Mathematica instructions at end of the fil

Post-CCSD(T) corrections to bond distances and vibrational frequencies: the power of Λ\Lambda

The importance of post-CCSD(T) corrections as high as CCSDTQ56 for ground-state spectroscopic constants ($D_e$, $\omega_e$, $\omega_ex_e$, and $\alpha_e$) has been surveyed for a sample of two dozen mostly heavy-atom diatomics spanning a broad range of static correlation strength. While CCSD(T) is known to be an unusually felicitous `Pauling point' between accuracy and computational cost, performance leaves something to be desired for molecules with strong static correlation. We find CCSDT(Q)$_\Lambda$ to be the next `sweet spot' up, of comparable or superior quality to the much more expensive CCSDTQ. A similar comparison applies to CCSDTQ(5)$_\Lambda$ vs. CCSDTQ5, while CCSDTQ5(6)$_\Lambda$ is essentially indistinguishable from CCSDTQ56. A composite of CCSD(T)-X2C/ACV5Z-X2C with [CCSDT(Q)$_\Lambda$ -- CCSD(T)]/cc-pVTZ or even cc-pVDZ basis sets appears highly effective for computational vibrational spectroscopy. Unlike CCSDT(Q) which breaks down for the ozone vibrational frequencies, CCSDT(Q)$_\Lambda$ handles them gracefully.Comment: Mol. Phys., in press [Timothy J. Lee memorial issue