A constraint on the thickness-weighted average equation of motion deduced from energetics

This study reviews the system governed by the thickness-weighted average (TWA) equation of motion, considering energetics. It is known that the TWA equation of motion based on the primitive equation describes the fluid motion with the residual mean velocity defined as the TWA velocity and is written in the same form as the nondissipative primitive equation, except that the eddy momentum fluxes (the interfacial form stress and Reynolds flux associated with eddy motion) are embedded in this equation. Also, incompressibility and density (buoyancy) conservation in the adiabatic condition hold in this system. In this study, considering that the TWA system satisfies a time mean energy conservation of the primitive equation system, we obtain an energy equation showing that the rate of change of eddy energies (the sum of the kinetic and potential energies of the eddies) along pathlines with the residual mean velocity is caused by the work done by the eddy momentum fluxes. This relation is analogous to the relation between internal energy and the dissipation function in a viscous fluid. This study also reconsiders the TWA system in terms of Hamiltonian dynamics. Regarding the eddy energies and the eddy momentum fluxes as analogous to the internal energy and the viscous momentum fluxes, respectively, the methodology of the variational principle for a viscous fluid can be applied to the TWA system. The Lagrangian density in this system is defined as the mean kinetic energy minus the mean potential energy and the eddy energies. Minimizing this Lagrangian density integrated over space and time under the constraints of the incompressibility equation, the buoyancy equation, and the equation of the eddy energies yields the TWA equation of motion. If we neglect the eddy energies in the Lagrangian density and the constraint of the equation of the eddy energies, the resulting equation in the variational calculus is merely the nondissipative primitive equation. This suggests that considering these is essential for describing the motion in the TWA system. Moreover, we inferred from the equation of the eddy energy that the TWA equation of motion can be expressed in a different form in which the isotropic component of the eddy momentum fluxes is included as a part of the pressure. Applying this modified equation to the issue of downstream decaying mechanism of the western boundary current extension jets, it can be interpreted that the deceleration of the jet is caused by the pressure induced by the eddies

Roles of axial anomaly on neutral quark matter with color superconducting phase

We investigate effects of the axial anomaly term with a chiral-diquark coupling on the phase diagram within a two-plus-one-flavor Nambu-Jona-Lasinio (NJL) model under the charge-neutrality and $\beta$-equilibrium constraints. We find that when such constraints are imposed, the new anomaly term plays a quite similar role as the vector interaction does on the phase diagram, which the present authors clarified in a previous work. Thus, there appear several types of phase structures with multiple critical points at low temperature $T$, although the phase diagrams with intermediate-$T$ critical point(s) are never realized without these constraints even within the same model Lagrangian. This drastic change is attributed to an enhanced interplay between the chiral and diquark condensates due to the anomaly term at finite temperature; the u-d diquark coupling is strengthened by the relatively large chiral condensate of the strange quark through the anomaly term, which in turn definitely leads to the abnormal behavior of the diquark condensate at finite $T$, inherent to the asymmetric quark matter. We note that the critical point from which the crossover region extends to zero temperature appears only when the strength of the vector interaction is larger than a critical value. We also show that the chromomagnetic instability of the neutral asymmetric homogenous two-flavor color superconducting(2CSC) phase is suppressed and can be even completely cured by the enhanced diquark coupling due to the anomaly term and/or by the vector interaction.Comment: 15 pages, 5 figures, typos corrected, new references and some statements adde

Renormalization-group Method for Reduction of Evolution Equations; invariant manifolds and envelopes

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set $t_0=t$ is naturally understood where $t_0$ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator $A$ in the evolution equation has no Jordan cell; when $A$ has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be the Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink-anti-kink and soliton-soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.Comment: 67 pages. No figures. v2: Additional discussions on the unstable motion in the the double-well potential are given in the text and the appendix added. Some references are also added. Introduction is somewhat reshape

QCD Phase Diagram: Phase Transition, Critical Point and Fluctuations

A summary of discussions on selected topics related to QCD phase diagram, phase transition, critical point, fluctuation and correlations at the Quark Matter 2009 conference are presented.Comment: Summary of the discussions on QCD Phase Diagram at 21st International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (QM2009), March 30 - 4 April, 2009, Knoxville, Tennessee, USA. New references adde

Lattice study of "f0_{0}(600) or σ\sigma"

We investigate the propagator of "f$_{0}$(600) or the $\sigma$" by the full-QCD simulation with Wilson fermions. We calculate the mesonic correlator in the I=0, $J^P=0^{+}$ channel on the $8^{3} \times 16$ lattice. Plaquet action and Wilson fermion action are adopted. A coupling constant $\beta$ is set to 4.8 and three kinds of hopping parameter, $\kappa$=0.1846, 0.1874 and 0.1891 are assayed. The disconnected diagram in the propagator is evaluated through taking average over 500 or 1000 Z2 noise. Simulations with the larger hopping parameter provide us with less noisy results. Though the statistics is not yet enough, our results indicate the existence of a pole with a mass in almost the same order as that of the $\rho$.Comment: Talk given at 20th International Symposium on Lattice Field Theory (LATTICE 2002), Boston, Massachusetts, 24-29 Jun 200

Lattice Study of Low-lying Nonet Scalar Mesons in Quenched Approximation

Using lattice QCD simulation in the quenched approximation, we study the $\kappa$ meson, which is ^3P_0 in the quark model, and compare experimental and other lattice data. The $\kappa$ is the lowest scalar meson with strangeness and constitutes the scalar nonet. The obtained mass is much higher than the recent experimental value, and therefore the $\kappa(800)$ is difficult to consider as a simple two-body constituent-quark structure, and may have another unconventional structure.Comment: 11pages, 5figure

Scalar Particles in Lattice QCD

We report a project to study scalar particles by lattice QCD simulations. After a brief introduction of the current situation of lattice study of the sigma meson, we describe our numerical simulations of scalar mesons, $\sigma$ and $\kappa$. We observe a low sigma mass, $m_\pi<m_\sigma\le m_\rho$, for which the disconnected diagram plays an important role. For the kappa meson, we obtain higher mass than the experimental value, i.e., $m_\kappa\sim 2m_{K^*}$.Comment: 4 figures, to be published in Proceedings of `International Symposium on Hadron Spectroscopy, Chiral Symmetry and Relativistic Description of Bound Systems' (in a series of KEK proceedings