Chihiro Matsuoka

Richtmyer-Meshkov instability: theory of linear and nonlinear evolution

A theoretical framework to study linear and nonlinear Richtmyer–Meshkov instability (RMI) is presented. This instability typically develops when an incident shock crosses a corrugated material interface separating two fluids with different thermodynamic properties. Because the contact surface is rippled, the transmitted and reflected wavefronts are also corrugated, and some circulation is generated at the material boundary. The velocity circulation is progressively modified by the sound wave field radiated by the wavefronts, and ripple growth at the contact surface reaches a constant asymptotic normal velocity when the shocks/rarefactions are distant enough. The instability growth is driven by two effects: an initial deposition of velocity circulation at the material interface by the corrugated shock fronts and its subsequent variation in time due to the sonic field of pressure perturbations radiated by the deformed shocks. First, an exact analytical model to determine the asymptotic linear growth rate is presented and its dependence on the governing parameters is briefly discussed. Instabilities referred to as RM-like, driven by localized non-uniform vorticity, also exist; they are either initially deposited or supplied by external sources. Ablative RMI and its stabilization mechanisms are discussed as an example. When the ripple amplitude increases and becomes comparable to the perturbation wavelength, the instability enters the nonlinear phase and the perturbation velocity starts to decrease. An analytical model to describe this second stage of instability evolution is presented within the limit of incompressible and irrotational fluids, based on the dynamics of the contact surface circulation. RMI in solids and liquids is also presented via molecular dynamics simulations for planar and cylindrical geometries, where we show the generation of vorticity even in viscid materials.

Magnetic Field Amplification Associated with the Richtmyer-Meshkov Instability

The amplification of a magnetic field due to the Richtmyer-Meshkov instability (RMI) is investigated by two-dimensional MHD simulations. Single-mode analysis is adopted to reveal definite relation between the nonlinear evolution of RMI and the field enhancement. It is found that an ambient magnetic field is stretched by fluid motions associated with the RMI, and the strength is amplified significantly by more than two orders of magnitude. The saturation level of the field is determined by a balance between the amplified magnetic pressure and the thermal pressure after shock passage. This effective amplification can be achieved in a wide range of the conditions for the RMI such as the Mach number of an incident shock and the density ratio at a contact discontinuity. The results suggest that the RMI could be a robust mechanism of the amplification of interstellar magnetic fields and cause the origin of localized strong fields observed at the shock of supernova remnants.

Notable effects of the metal salts on the formation and decay reactions of α-tocopheroxyl radical in acetonitrile solution. The complex formation between α-tocopheroxyl and metal cations.

The measurement of the UV-vis absorption spectrum of α-tocopheroxyl (α-Toc(•)) radical was performed by reacting aroxyl (ArO(•)) radical with α-tocopherol (α-TocH) in acetonitrile solution including four kinds of alkali and alkaline earth metal salts (MX or MX(2)) (LiClO(4), LiI, NaClO(4), and Mg(ClO(4))(2)), using stopped-flow spectrophotometry. The maximum wavelength (λ(max)) of the absorption spectrum of the α-Toc(•) at 425.0 nm increased with increasing concentration of metal salts (0-0.500 M) in acetonitrile, and it approached constant values, suggesting an [α-Toc(•)-M(+) (or M(2+))] complex formation. The stability constants (K) were determined to be 9.2, 2.8, and 45 M(-1) for LiClO(4), NaClO(4), and Mg(ClO(4))(2), respectively. By reacting ArO(•) with α-TocH in acetonitrile, the absorption of ArO(•) disappeared rapidly, while that of α-Toc(•) appeared and then decreased gradually as a result of the bimolecular self-reaction of α-Toc(•) after passing through the maximum. The second-order rate constants (k(s)) obtained for the reaction of α-TocH with ArO(•) increased linearly with an increasing concentration of metal salts. The results indicate that the hydrogen transfer reaction of α-TocH proceeds via an electron transfer intermediate from α-TocH to ArO(•) radicals followed by proton transfer. Both the coordination of metal cations to the one-electron reduced anions of ArO(•) (ArO:(-)) and the coordination of counteranions to the one-electron oxidized cations of α-TocH (α-TocH(•)(+)) may stabilize the intermediate, resulting in the acceleration of electron transfer. A remarkable effect of metal salts on the rate of bimolecular self-reaction (2k(d)) of the α-Toc(•) radical was also observed. The rate constant (2k(d)) decreased rapidly with increasing concentrations of the metal salts. The 2k(d) value decreased at the same concentration of the metal salts in the following order: no metal salt > NaClO(4) > LiClO(4) > Mg(ClO(4))(2). The complex formation between α-Toc(•) and metal cations may stabilize the energy level of the reactants (α-Toc(•) + α-Toc(•)), resulting in the decrease of the rate constant (2k(d)). The alkali and alkaline earth metal salts having a smaller ionic radius of cation and a larger charge of cation gave larger K and k(s) values and a smaller 2k(d) value.

Vortex sheet motion in incompressible Richtmyer–Meshkov and Rayleigh–Taylor instabilities with surface tension

Motion of a planar interface in incompressible Richtmyer–Meshkov (RM) and Rayleigh–Taylor (RT) instabilities with surface tension is investigated numerically by using the boundary integral method. It is shown that when the Atwood number is small, an interface rolls up without regularization of the interfacial velocity. A phenomenon known as “pinching” in the physics of drops is observed in the final stage of calculations at various Atwood numbers and surface tension coefficients, and it is shown that this phenomenon is caused by a vortex dipole induced on the interface. It is also shown that when the surface tension coefficient is large, finite amplitude standing wave solutions exist for the RM instability. This standing wave solution is investigated in detail by nonlinear stability analysis. When gravity is taken into account (RT instability), linearly stable but nonlinearly unstable motion can occur under a critical condition that the frequency of the linear dispersion relation in the system is equal to zero. Further, it is shown that the growth rate of bubbles and spikes under this critical motion is neither of the exponential type nor of the power law type at both the linear stage and the asymptotic stage.

Numerical studies on scattering of the NLS soliton due to an impurity

Abstract The scattering of nonlinear Schrodinger solitons due to an impurity in anharmonic lattices is studied numerically. At most one soliton is generated in both reflected and transmitted waves. Their amplitudes coincide very well with those of the theoretical result, which has been obtained through the inverse scattering method.

Simulation of Envelope Soliton Scattering in Discontinuous Media

Scattering of the nonlinear Schrodinger (NLS) solitons in discontinuous media is studied numerically. As a physical model, one-dimensional anharmonic lattice which has a discontinuity in its mass distribution, is analized. After the collision of the incident NLS soliton against the mass interface, we detect the amplitude of the reflected and transmitted solitons. They coincide very well with those of the theoretcal ones (T. Iizuka and M. Wadati, J. Phys. Soc. Jpn. 61 (1992) 3077), which is based on the inverse scattering method.

Three-dimensional vortex sheet motion with axial symmetry in incompressible Richtmyer–Meshkov and Rayleigh–Taylor instabilities

The motion of vortex sheets in three-dimensional inviscid and incompressible Richtmyer–Meshkov and Rayleigh–Taylor instabilities (RMI and RTI) with axial symmetry is investigated analytically and numerically. A comparison is made with the homogeneous case where density stratification does not exist, and possible solutions are discussed for RMI and RTI. When density stratification exists, it is shown that azimuthal motion (deformation) of the vortex sheet occurs without a swirl flow. This azimuthal motion is different from the rotational motion by the swirl. We present the numerical results for the interfacial motion in RMI and RTI with axial symmetry obtain by the vortex blob method, and discuss the differences from the motion of a vortex sheet in a homogeneous fluid.

Vortex Dynamics of the Complex Ginzburg-Landau Equation

Equations of motion of vortices of the complex Ginzburg-Landau equation are derived using matched asymptotic expansions. It is shown that two vortices are repulsive and attractive for equal and opposite charges, respectively.

Computation of entropy and Lyapunov exponent by a shift transform.

We present a novel computational method to estimate the topological entropy and Lyapunov exponent of nonlinear maps using a shift transform. Unlike the computation of periodic orbits or the symbolic dynamical approach by the Markov partition, the method presented here does not require any special techniques in computational and mathematical fields to calculate these quantities. In spite of its simplicity, our method can accurately capture not only the chaotic region but also the non-chaotic region (window region) such that it is important physically but the (Lebesgue) measure zero and usually hard to calculate or observe. Furthermore, it is shown that the Kolmogorov-Sinai entropy of the Sinai-Ruelle-Bowen measure (the physical measure) coincides with the topological entropy.

Multi-mode character of the nonlinear dynamics of a vortex sheet in Rayleigh-Taylor and Richtmyer-Meshkov instabilities

Multi-modal character of interfaces in Rayleigh-Taylor (RT) and incompressible Richtmyer-Meshkov (RM) instabilities with planar and cylindrical geometries is studied numerically. An interface is treated as a vortex sheet and the vortex method which regularizes the singularity caused by the Cauchy integral in equation of motion is adapted for the calculations of interfacial motion. Successive profiles of interfaces and the temporal evolution of the strength of a vortex sheet are presented, and discussed the differences between pure cosine or sine mode and the mixed mode which includes both cosine and sine modes in the initial disturbance. It is shown that the sheet strength of the interface with multi-mode is considerably larger than that with single-mode for the RM instability, while the difference between those two modes is almost not found for the RT instability.