1. |
Yasuhide Fukumoto, Rong Zou, Nambu bracket for 3D ideal fluid dynamics and magnetohydrodynamics, *Prog. Theor. Exp. Phys. (PTEP) *, https://doi.org/10.1093/ptep/ptae025, 2024, ptae025-1-ptae025-2, 2024.02, The ideal magnetohydrodynamics (MHD) as well as the ideal fluid dynamics is governed by the Hamilton equation with respect to the Lie-Poisson bracket. The Nambu bracket manifestly represents the Lie-Poisson structure in terms of derivative of the Casimir invariants. We construct a compact Nambu-bracket representation for the three-dimensional ideal MHD equations, with use of three Casimirs for the second Hamiltonians, the total entropy and the magnetic and cross helicities, whose coefficients are all constant. The Lie-Poisson bracket induced by this Nambu bracket does not coincide with the original one, but supplemented by terms with an auxiliary variable. The supplemented Lie-Poisson bracket qualifies the cross-helicity as the Casimir. By appealing to Noether's theorem, we show that the cross-helicity is the integral invariant associated with the particle-relabeling symmetry. Employing the Lagrange label function, as the independent variable in the variational framework, facilitates implementation of the relabeling transformation. By incorporating the divergence symmetry, other known topological invariants are put on the same ground of Noether's theorem.. |

2. |
Linh Thi Hoai Nguyen, Yasuhide Fukumoto, Pierluigi Cesana, Aleksandar Staykov, Fully automatized optimization of ring-opening reactions in lactone derivatives via 2-step machine learning, *Journal of Physical Chemistry A*, https://doi.org/10.1021/acs.jpca.3c05887, 1-7, 2023.11. |

3. |
Yasuhide Fukumoto, Rong Zou, Isomagnetovortical perturbations and wave energy of MHD flows, *Reviews of Modern Plasma Physics*, 10.1007/s41614-023-00113-8, 7, 8-1-8-21, 2023.06. |

4. |
Kazuo Matsuura, Yasuhide Fukumoto, Hierarchical clustering method of volumetric vortical regions with application to the late-stage of laminar-turbulent transition, *Physical Review Fluids*, https://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.7.054703, 7, 054703-46, 2022.06. |

5. |
D. Packwood, L.T.H. Nguyen, P. PCesana, G. Zhang, A. Staykov, Yasuhide Fukumoto, D.H. Nguyen, Machine learning in materials chemistry: An invitation, *Machine Learning with Application*, 10.1016/j.mlwa.2022.100265, 8, 100265-1-100265-31, 2022.06. |

6. |
T.T. Le, Yasuhide Fukumoto, Effect of depth discontinuity on interfacial stability of tangential-velocity discontinuity in shallow-water flow, *PHYSICS LETTERS A*, 10.1016/j.physleta.2022.128073, 436, 128073-1-128073-7, 2022.06. |

7. |
V. L. Okulov, Yasuhide Fukumoto, Review of Analytical Approaches for Simulating Motions of Helical Vortex, *FRONTIERS IN ENERGY RESEARCH*, 10.3389/fenrg.2022.817941, 10, 817941-1-817941-6, 2022.02. |

8. |
K. Kuwana, K. Matsue, Yasuhide Fukumoto, R. Dobashi, K. Saito, Fire whirls: A Combustion Science Perspective, *COMBUSTION SCIENCE AND TECHNOLOGY*, 10.1080/00102202.2021.2019234, 2022.01. |

9. |
X. Zhao, F. Liu, Yasuhide Fukumoto, A multi level linearized Crank-Nicolson scheme for Richards equation under variable flux boundary conditions, *Applicable Analysis*, 10.1080/00036811.2021.1992395, 2021.10. |

10. |
V. L. Okulov, Yasuhide Fukumoto, Singular approximation for calculation of vortex filaments, *Journal of Applied Mechanics and Technical Physics*, 10.1134/S0021894421030196, 62, 519-524, 2021.09. |

11. |
Mako Sato, Yasuhide Fukumoto, Influence of an oblique magnetic field on planar flame front instability, * 2019-20 MATRIX Annals*, 10.1007/978-3-030-62497-2_26, 439-459, 2020.10. |

12. |
Keigo Wada, Yasuhide Fukumoto, Compressibility effect on Markstein number for a flame front in long-wavelength approximation, *2019-20 MATRIX Annals*, 10.1007/978-3-030-62497-2_19, 329-350, 2020.10. |

13. |
V. L. Okulov, Yasuhide Fukumoto, Analytical solution for self-induced motion of a helical vortex with a Gaussian core, *Thermophysics and Aeromechanics*, 10.1134/S0869864320040022, 27, 4, 481-488, 2020.12, The paper presents an analytical solution for helical vortices with a Gaussian vorticity distribution in the core, which is confirmed by experimental and numerical simulations. This result is obtained by extending the Dyson method to the Biot–Savart law. Previously, analytical solutions were found and studied only for vortices with constant vorticity distribution in the core (a Rankine-type vortex core). One of the important issues raised during the discussion is the difference between self-induced movements of helical structures with both types of vortex core. The proposed solutions are important for the fundamental understanding and description of the behavior of helical eddy flows in various fields of industry and in nature. Examples include tip vortices behind the rotors of wind or hydro turbines, tornadoes, or axial vortices in aerodynamic devices such as vortex apparatuses and generators; cyclone separators, combustion chambers, etc.. |

14. |
A. B. Samokhin, Yasuhide Fukumoto, Singular modes of the integral scattering operator in anisotropic inhomogeneous media, *Differential Equations*, 10.1134/S0012266120090104, 56, 9, 1212-1218, 2020.10, We study singular modes of volume singular integral equations describing problems of scattering of electromagnetic wave in anisotropic dielectric structures. An explicit form of these modes is obtained for a certain class of anisotropic media. Example of real media in which singular modes can exist are considered.. |

15. |
Jishan Fan, Yashuhide Fukumoto,Yong Zhou, Regularity criteria for a Ginzburg-Landau-Navier-Stokes in superfluidity in R^n, *Mathematical Methods in the Applied Sciences*, 10.1002/mma.6397, 43, 10, 6542-6552, 2020.04, In this work, we prove some regularity criteria for a Ginzburg-Landau-Navier-Stokes in superfluidity in ℝ𝑛(𝑛≥3) .. |

16. |
Liu Fengnan, Yasuhide Fukumoto, Xiaopeng Zhao, A Linearized Finite Difference Scheme for the Richards Equation Under Variable-Flux Boundary Conditions, *Journal of Scientific Computing*, 10.1007/s10915-020-01196-y, 83, 1, 2020.04, The Richards equation is a degenerate nonlinear PDE that models a flow through saturated/unsaturated porous media. Research on its numerical methods has been conducted in many fields. Implicit schemes based on a backward Euler format are widely used in calculating it. However, it is difficult to obtain stability with a numerical scheme because of the strong nonlinearity and degeneracy. In this paper, we establish a linearized semi-implicit finite difference scheme that is faster than backward Euler implicit schemes. We analyze the stability of this scheme by adding a small positive perturbation ϵ to the coefficient function of the Richards equation. Moreover, we show that there is a linear relationship between the discretization error in the L^{∞}-norm and ϵ. Numerical experiments are carried out to verify our main results.. |

17. |
Fengnan Liu, Yasuhide Fukumoto, Xiaopeng Zhao, Stability analysis of the explicit difference scheme for richards equation, *Entropy*, 10.3390/e22030352, 22, 3, 2020.03, A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In addition, we introduce an extra term in the difference scheme which is used to relax the time step restriction for improving the stability condition. With the augmented terms, we prove the stability using the induction method. Numerical experiments show the validity and the accuracy of the scheme, along with its efficiency.. |

18. |
R. Zou, J. Labarbe, O. N. Kirillov, Y. Fukumoto, Analysis of azimuthal magnetorotational instability of rotating magnetohydrodynamic flows and Tayler instability via an extended Hain-Lüst equation, *Physical Review E*, 10.1103/PhysRevE.101.013201, 101, 1, 2020.01, We consider a differentially rotating flow of an incompressible electrically conducting and viscous fluid subject to an external axial magnetic field and to an azimuthal magnetic field that is allowed to be generated by a combination of an axial electric current external to the fluid and electrical currents in the fluid itself. In this setting we derive an extended version of the celebrated Hain-Lüst differential equation for the radial Lagrangian displacement that incorporates the effects of the axial and azimuthal magnetic fields, differential rotation, viscosity, and electrical resistivity. We apply the Wentzel-Kramers-Brillouin method to the extended Hain-Lüst equation and derive a comprehensive dispersion relation for the local stability analysis of the flow to three-dimensional disturbances. We confirm that in the limit of low magnetic Prandtl numbers, in which the ratio of the viscosity to the magnetic diffusivity is vanishing, the rotating flows with radial distributions of the angular velocity beyond the Liu limit, become unstable subject to a wide variety of the azimuthal magnetic fields, and so is the Keplerian flow. In the analysis of the dispersion relation we find evidence of a new long-wavelength instability which is caught also by the numerical solution of the boundary value problem for a magnetized Taylor-Couette flow.. |

19. |
Jishan Fan, Yasuhide Fukumoto, Yong Zhou, Regularity criteria for a Ginzburg-Landau-Navier-Stokes in superfluidity in Rn, *Mathematical Methods in the Applied Sciences*, 10.1002/mma.6397, 43, 10, 6542-6552, 2020.01, In this work, we prove some regularity criteria for a Ginzburg-Landau-Navier-Stokes in superfluidity in (Formula presented.).. |

20. |
Ning Duan, Yasuhide Fukumoto, Xiaopeng Zhao, Asymptotic behavior of solutions to incompressible electron inertial Hall-MHD system in R^{3}, *Communications on Pure and Applied Analysis*, 10.3934/cpaa.2019136, 18, 6, 3035-3057, 2019.11, In this paper, by using Fourier splitting method and the properties of decay character r^{*}, we consider the decay rate on higher order derivative of solutions to 3D incompressible electron inertial Hall-MHD system in Sobolev space H^{s}(R^{3}) × H^{s} ^{+1}(R^{3}) for s ∈ N^{+}. Moreover, based on a parabolic interpolation inequality, bootstrap argument and some weighted estimates, we also address the space-time decay properties of strong solutions in R^{3}. |

21. |
Liangbing Jin, Lê Thị Thái, Yasuhide Fukumoto, Frictional effect on stability of discontinuity interface in tangential velocity of a shallow-water flow, *Physics Letters, Section A: General, Atomic and Solid State Physics*, 10.1016/j.physleta.2019.125839, 383, 26, 2019.09, We examine a frictional effect on the linear stability of an interface of discontinuity in tangential velocity. The fluid is moving with uniform velocity U in a region but is at rest in the other, and the bottom surface is assumed to exert drag force, quadratic in velocity, on the thin fluid layer. In the absence of the drag, the instability of the Kelvin-Helmholtz type is suppressed for U>8c, with c being the propagating speed of the gravity wave. We find by asymptotic analyses for both small and large values of the drag strength that the drag, regardless of its strength, makes the flow unstable for the whole range of the Froude number U/c.. |

22. |
Yasuhide Fukumoto, Xiaopeng Zhao, Well-posedness and large time behavior of solutions for the electron inertial Hall-MHD system, *Advances in Differential Equations*, 24, 1-2, 31-68, 2019.01, In this paper, the properties of weak and strong solutions for the Hall-magnetohydrodynamic system augmented by the effect of elec- tron inertia are studied. First, we establish the existence and uniqueness of local-in-time strong solutions; Then, we prove the existence of global strong solutions under the condition that ∥u _{0} ∥ _{H˙ 1/2} +∥B _{0} ∥ _{H˙ 1/2} +∥∇B _{0} ∥ _{H˙ 1/2} is sufficiently small. Moreover, by applying a cut-off function and gen- eralized energy inequality, we show that the weak solution of electron inertia Hall-MHD system approaches zero as the time t → ∞. Finally, the algebraic decay rate of the weak solution of electron inertia Hall- MHD system is established by using Fourier splitting method and the properties of decay character.. |

23. |
D. V. Ilyin, Y. Fukumoto, W. A. Goddard, S. I. Abarzhi, Analysis of dynamics, stability, and flow fields' structure of an accelerated hydrodynamic discontinuity with interfacial mass flux by a general matrix method, *Physics of Plasmas*, 10.1063/1.5008648, 25, 11, 2018.11, We develop a general matrix method to analyze from a far field the dynamics of an accelerated interface between incompressible ideal fluids of different densities with interfacial mass flux and with negligible density variations and stratification. We rigorously solve the linearized boundary value problem for the dynamics conserving mass, momentum, and energy in the bulk and at the interface. We find a new hydrodynamic instability that develops only when the acceleration magnitude exceeds a threshold. This critical threshold value depends on the magnitudes of the steady velocities of the fluids, the ratio of their densities, and the wavelength of the initial perturbation. The flow has potential velocity fields in the fluid bulk and is shear-free at the interface. The interface stability is set by the interplay of inertia and gravity. For weak acceleration, inertial effects dominate, and the flow fields experience stable oscillations. For strong acceleration, gravity effects dominate, and the dynamics is unstable. For strong accelerations, this new hydrodynamic instability grows faster than accelerated Landau-Darrieus and Rayleigh-Taylor instabilities. For given values of the fluids' densities and their steady bulk velocities, and for a given magnitude of acceleration, we find the critical and maximum values of the initial perturbation wavelength at which this new instability can be stabilized and at which its growth is the fastest. The quantitative, qualitative, and formal properties of the accelerated conservative dynamics depart from those of accelerated Landau-Darrieus and Rayleigh-Taylor dynamics. New diagnostic benchmarks are identified for experiments and simulations of unstable interfaces.. |

24. |
Ummu Habibah, Hironori Nakagawa, Yasuhide Fukumoto, Finite-thickness effect on speed of a counter-rotating vortex pair at high Reynolds numbers, *Fluid Dynamics Research*, 10.1088/1873-7005/aaa5c8, 50, 3, 2018.03, We establish a general formula for the translational speed of a counter-rotating vortex pair, valid for thick cores, moving in an incompressible fluid with and without viscosity. We extend to higher order the method of matched asymptotic expansions developed by Ting and Tung (1965 Phys. Fluids 8 1039-51). The solution of the Euler or the Navier-Stokes equations is constructed in the form of a power series in a small parameter, the ratio of the core radius to the distance between the core centers. For a viscous vortex pair, the small parameter should be where ν is the kinematic viscosity of the fluid and Γ is the circulation of each vortex. A correction due to the effect of finite thickness of the vortices to the traveling speed makes its appearance at fifth order. A drastic simplification is achieved of expressing it solely in terms of the strength of the second-order quadrupole field associated with the elliptical deformation of the core. For a viscous vortex pair, we exploit the conservation law for the hydrodynamic impulse to derive the growth of the distance between the vortices, which is cubic in time.. |

25. |
Fermín Franco Medrano, Yasuhide Fukumoto, Clara Marika Velte, Azur Hodžić, Mass entrainment rate of an ideal momentum turbulent round jet, *journal of the physical society of japan*, 10.7566/JPSJ.86.034401, 86, 3, 2017.01, We propose a two-phase-fluid model for a full-cone turbulent round jet that describes its dynamics in a simple but comprehensive manner with only the apex angle of the cone being a disposable parameter. The basic assumptions are that (i) the jet is statistically stationary and that (ii) it can be approximated by a mixture of two fluids with their phases in dynamic equilibrium. To derive the model, we impose conservation of the initial volume and total momentum fluxes. Our model equations admit analytical solutions for the composite density and velocity of the two-phase fluid, both as functions of the distance from the nozzle, from which the dynamic pressure and the mass entrainment rate are calculated. Assuming a far-field approximation, we theoretically derive a constant entrainment rate coefficient solely in terms of the cone angle. Moreover, we carry out experiments for a single-phase turbulent air jet and show that the predictions of our model compare well with this and other experimental data of atomizing liquid jets.. |

26. |
Hemanta Hazarika, Yasuhide Fukumoto, Sustainable solution for seawall protection against tsunami-induced damage, *International Journal of Geomechanics*, 10.1061/(ASCE)GM.1943-5622.0000687, 16, 5, 2016.10, To protect coastal structures from the damage caused by the impact force of a tsunami, a new concept of using waste tires behind such structures is introduced in this paper. A physical model for tsunami impact force simulation was developed to evaluate the reduction effect of tsunami impact force by the tire structures. Model tests also were performed to evaluate the stiffness of tire structures. From an esthetic point of view, cultivation of suitable plants inside the tires is also proposed. Field tests on planting trees that can grow in saline soil conditions were performed to see whether such a structure can preserve the greenery of the area. Results show that the tsunami impact force could be reduced considerably by placing filled tires (with a suitable material) behind seawalls, and this technique can protect the structures from the tsunami impact force and the resulting scouring. The greening effect could be maintained by the appropriate selection of the shrubs and trees planted inside the tires, making it one of the most cost-effective methods for recycling waste tires.. |

27. |
Michael I. Tribelsky, Yasuhide Fukumoto, Laser heating of dielectric particles for medical and biological applications, *Biomedical Optics Express*, 10.1364/BOE.7.002781, 7, 7, 2781-2788, 2016.07, We consider the general problem of laser pulse heating of a spherical dielectric particle embedded in a liquid. The discussed range of the problem parameters is typical for medical and biological applications. We focus on the case, when the heat diffusivity in the particle is of the same order of magnitude as that in the fluid. We perform quantitative analysis of the heat transfer equation based on interplay of four characteristic scales of the problem, namely the particle radius, the characteristic depth of light absorption in the material of the particle and the two heat diffusion lengths: in the particle and in the embedding liquid. A new quantitative characteristic of the laser action, that is the cooling time, describing the temporal scale of the cooling down of the particle after the laser pulse is over, is introduced and discussed. Simple analytical formulas for the temperature rise in the center of the particle and at its surface as well as for the cooling time are obtained. We show that at the appropriate choice of the problem parameters the cooling time may be by many orders of magnitude larger the laser pulse duration. It makes possible to minimize the undesirable damage of healthy tissues owing to the finite size of the laser beam and scattering of the laser radiation, simultaneously keeping the total hyperthermia period large enough to kill the pathogenic cells. An example of application of the developed approach to optimization of the therapeutic effect at the laser heating of particles for cancer therapy is presented.. |

28. |
Yasuhide Fukumoto, Valery L. Okulov, David H. Wood, The Contribution of Kawada to the Analytical Solution for the Velocity Induced by a Helical Vortex Filament, *Applied Mechanics Reviews*, 10.1115/1.4031964, 67, 6, 2015.11, The basic solution for the velocity induced by helical vortex filament is well known as Hardin's solution, published in 1982. A study of early publications on helical vortices now shows that the Japanese scientist Kawada from Tokyo Imperial University also produced many of these results in 1936, which predates Hardin by 46 years. Consequently, in order to honor both, we have studied their derivations to establish the originality of both solutions.. |

29. |
Yasuhide Fukumoto, Youichi Mie, Lagrangian approach to weakly nonlinear interaction of Kelvin waves and a symmetry-breaking bifurcation of a rotating flow, *Fluid Dynamics Research*, 10.1088/0169-5983/47/1/015509, 47, 1, 015509 (15pp), 2015.02. |

30. |
Snezhana I Abarzhi, Yasuhide Fukumoto, Leo P Kadanoff, Stability of a hydrodynamic discontinuity, *Physica Scripta*, 10.1088/0031-8949/90/1/018002, 90, 018002 (7pp), 2015.01. |

31. |
Jishan Fan, Ahmed Alsaedi, Yasuhide Fukumoto, Tasawar Hayat, Yong Zhou, A regularity criterion for the density-dependent hall-magnetohydrodynamics, *Zeitschrift fur Analysis und ihre Anwendung*, 10.4171/ZAA/1539, 34, 3, 277-284, 2015.01, This paper proves a regularity criterion for the density-dependent Hallmagnetohydrodynamics with positive density.. |

32. |
Hirofumi Sakuma, Yasuhide Fukumoto, On formal stability of stratified shear flows, *Publications of the Research Institute for Mathematical Sciences*, 10.4171/PRIMS/166, 51, 4, 605-633, 2015.01, A novel linear stability criterion is established for the equilibria of general three-dimen- sional (3D) rotating ows of an ideal gas satisfying Boyle–Charles' law by a newly refined energy-Casimir convexity (ECC) method that can exploit a larger class of Casimir in- variants. As the conventional ECC method cannot be applied directly to stratified shear ows, in our new approach, rather than checking the local convexity of a Lyapunov func- tional L ≡ E + C_{E} defined as a sum of the total energy and a certain Casimir, we seek the condition for nonexistence of unstable manifolds: orbits (physically realisable ow in phase space) on the leaves of invariants including L as well as other Casimirs connecting a given equilibrium point O and other points in the neighbourhood of it. We argue that the separatrices of the second variation of L (δ^{2}L = 0) generally consist of such unstable manifolds as well as pseudo unstable ones for which either the total energy or Casimirs actually serve as a barrier for escaping orbits. The significance of the new method lies in the fact that it eliminates the latter so as to derive a condition for O being an isolated equilibrium point in terms of orbital connections.. |

33. |
Y Fukumoto, M Miyajima, The localized induction hierarchy and the Lund-Regge equation, *JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL*, 10.1088/0305-4470/29/24/025, 29, 24, 8025-8034, 1996.12, An evolution equation of a curve is constructed by summing up the infinite sequence of commuting vector fields of the integrable hierarchy for the localized induction equation (LIE). It is shown to be equivalent to the Lund-Regge equation. The intrinsic equations governing the curvature and torsion are deduced in the form of integrodifferential evolution equations. A class of exact solutions which correspond to the permanent forms of a curve evolving by a steady rigid motion are presented. The analysis of the solutions reveals that, given the shape, there are two speeds of motion, one of which has no counterpart in the case of the LIE.. |

34. |
Y Fukumoto, T Miyazaki, Local stability of two-dimensional steady irrotational solenoidal flows with closed streamlines, *JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN*, 10.1143/JPSJ.65.107, 65, 1, 107-113, 1996.01, We investigate the linear stability of regions of two-dimensional steady irrotational flows in which streamlines are closed, with respect to three-dimensional perturbations of short wavelengths. The fluid is assumed to be inviscid and incompressible. The geometrical optics equations derived by Hameiri and Lifschitz are employed. It is demonstrated that, at large times, both short-wave perturbation velocity and vorticity grow in magnitude at most linearly in time. It implies that there are no exponentially growing short-wave instabilities in such a flow domain. The same result holds if the how field is steady relative to a suitably chosen steadily rotating frame and the net vorticity relative to the inertial frame is zero.. |

35. |
Y Fukumoto, Stationary configurations of a vortex filament in background flows, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, 10.1098/rspa.1997.0067, 453, 1961, 1205-1232, 1997.06, We investigate the three-dimensional configurations of a thin vortex filament embedded in background flows of an inviscid incompressible fluid, based on the localized induction approximation. An analogy is found between stationary configurations of a vortex filament in a steady flow and trajectories of a charged particle in a steady magnetic field. An analogy with trajectories of a sound ray in a steady low-Mach number flow is drawn as well. These analogies allow us to use the Lagrangian and Hamiltonian formalism of classical mechanics for calculating fully nonlinear forms of a vortex filament. The equations for the filament curve are integrable when the background flow has two spatial symmetries. Only integrable cases are explored in some detail. To illustrate the advantages of use of the analogy, we re-examine the invariant shapes of a vortex filament moving through a still fluid obtained by Kida (1981). Analogies of the Kida class with the motions of a heavy symmetrical top and a charged spherical pendulum in the field of a magnetic monopole are discussed. As next examples, we consider a point source (sink) and a line source (sink) flows. A filament takes the form of a non-uniform helix wound around a cone. The last example is a circular jet of a parabolic velocity profile. We construct a closed filament with azimuthal waves, lying on a plane orthogonal to the flow direction. Among them, an elliptic vortex ring is found to be distinct from higher waves.. |

36. |
T Rozi, Y Fukumoto, The most unstable perturbation of wave-packet form inside Hill's vortex, *JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN*, 10.1143/JPSJ.69.2700, 69, 8, 2700-2701, 2000.08. |

37. |
Y Fukumoto, HK Moffatt, Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity, *JOURNAL OF FLUID MECHANICS*, 10.1017/S0022112000008995, 417, 1-45, 2000.08, A large-Reynolds-number asymptotic solution of the Navier-Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. In order to take account of the influence of elliptical deformation of the core due to the self-induced strain, the method of matched asymptotic expansions is extended to a higher order in a small parameter epsilon = (nu/Gamma)(1/2), where nu is the kinematic viscosity of fluid and Gamma is the circulation. Alternatively, epsilon is regarded as a measure of the ratio of the core radius to the ring radius, and our scheme is applicable also to the steady inviscid dynamics. We establish a general formula for the translation speed of the ring valid up to third order in epsilon. This is a natural extension of Fraenkel-Saffman's first-order formula, and reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an infinitely thin circular loop of radius R-0, that viscosity acts, at third order, to expand the circles of stagnation points of radii R-s(t) and (R) over tilde(s)(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius R-p(t) as R-s approximate to R-0 + [2 log(4R(0)/root nu t) + 1.4743424] nu t/R-0, (R) over tilde(s) approximate to R-0 + 2.5902739 nu t/R-0, and R-p approximate to R-0 + 4.5902739 nu t/R-0. The growth of the radial centroid of vorticity, linear in time, is also deduced. The results are compatible with the experimental results of Sallet & Widmayer (1974) and Weigand & Gharib (1997). The procedure of pursuing the higher-order asymptotics provides a clear picture of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a line of dipoles along the core centreline, with their axes in the propagating direction, subjected to the self-induced flow field. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specified at the initial instant. This specification removes an indeterminacy of the first-order theory. We derive a new asymptotic development of the Biot-Savart law for an arbitrary distribution of vorticity, which makes the non-local induction velocity from the dipoles calculable at third order.. |

38. |
Asymptotic expansions for the motion of a curved vortex filament and the localized induction hierarchy (Mathematical Analysis in Fluid and Gas Dynamics). |

39. |
Y Fukumoto, Three-dimensional motion of a vortex filament and its relation to the localized induction hierarchy, *EUROPEAN PHYSICAL JOURNAL B*, 10.1140/epjb/e2002-00279-5, 29, 2, 167-171, 2002.09, Three-dimensional motion of a slender vortex tube, embedded in an inviscid incompressible fluid, is investigated under the localized induction approximation for the Euler equations. Using the method of matched asymptotic expansions in a small parameter c, the ratio of core radius to curvature radius, the velocity of a vortex filament is derived to O(epsilon(3)), whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account. It is found that there is an integrable line in the core whose evolution obeys a summation of the first and third terms of the localized induction hierarchy.. |

40. |
Y Fukumoto, Higher-order asymptotic theory for the velocity field induced by an inviscid vortex ring, *FLUID DYNAMICS RESEARCH*, 10.1016/S0169-5983(01)00044-2, 30, 2, 65-92, 2002.02, We explore the flow field around a thin axisymmetric vortex ring steadily translating in an ideal fluid, with vorticity proportional to distance from the axis of symmetry, originally treated by Dyson (Pihilos. Trans. R. Soc. 184 (1893) 1041). By making a higher-order extension of the method of matched asymptotic expansions in a small. parameter epsilon, the ratio of the core radius to the ring radius R-0, an explicit form of the streamfunction is derived, to O(epsilon(3)), both inside and outside the core. The pressure and velocity fields are written out in full to the same order. It is shown that the circle of minimum pressure coincides, to O(epsilon(2)R(0)), with the circle of stagnation points viewed from the frame moving with the core. (C) 2002 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.. |

41. |
Y Fukumoto, The three-dimensional instability of a strained vortex tube revisited, *JOURNAL OF FLUID MECHANICS*, 10.1017/S0022112003006025, 493, 287-318, 2003.10, We revisit the Moore-Saffman-Tsai-Widnall instability, a parametric resonance between left- and right-handed bending waves of infinitesimal amplitude, on the Rankine vortex strained by a weak pure shear flow. The results of Tsai & Widnall (1976) and Eloy & Le Dizes (2001), as generalized to all pairs of Kelvin waves whose azimuthal wavenumbers m are separated by 2, are simplified by providing an explicit solution of the linearized Euler equations for the disturbance flow field. Given the wavenumber k(0) and the frequency coo of an intersection point of dispersion curves, the growth rate is expressible solely in terms of the modified Bessel functions, and so is the unstable wavenumber range. Every intersection leads to instability. Most of the intersections correspond to weak instability that vanishes in the short-wave limit, and dominant modes are restricted to particular intersections. For helical waves m = +/-1, the growth rate of non-rotating waves is far larger than that of rotating waves. The wavenumber width of stationary instability bands broadens linearly in k(0), while that of rotating instability bands is bounded. The growth rate of the stationary instability takes, in the long-wavelength limit, the value of epsilon/2 for the two-dimensional displacement instability, and, in the short-wavelength limit, the value of 9epsilon/16 for the elliptical instability, being larger at large but finite values of k(0). Here E is the strength of shear near the core centre. For resonance between higher azimuthal wavenumbers m and m + 2, the same limiting value is approached as ko --> infinity, along sequences of specific crossing points whose frequency rapidly converges to m + 1, in two ways, from above for a fixed m and from below for m --> infinity. The energy of the Kelvin waves is calculated by invoking Cairns' formula. The instability result is compatible with Krein's theory for Hamiltonian spectra.. |

42. |
Y Hattori, Y Fukumoto, Short-wavelength stability analysis of thin vortex rings, *PHYSICS OF FLUIDS*, 10.1063/1.1606446, 15, 10, 3151-3163, 2003.10, The linear stability of thin vortex rings are studied by short-wavelength stability analysis. The modified Hill-Schrodinger equation for vortex rings, which incorporates curvature effect, is derived. It is used to evaluate growth rates analytically. The growth rates are also evaluated by numerical calculation and they agree well with analytical values for small epsilon which is the ratio of core radius to ring radius. Two types of vortex rings are considered: Kelvin's vortex ring and a Gaussian vortex ring. For Kelvin's vortex ring the maximum first-order growth rate is found to be 165/256 epsilon. For the Gaussian vortex ring the first-order growth rate is large in the skirts of the vortex core. The first-order instability is significant for both vortex rings. (C) 2003 American Institute of Physics.. |

43. |
Y Fukumoto, S Lugomer, Instability of vortex filaments in laser-matter interactions, *PHYSICS LETTERS A*, 10.1016/S0375-9601(03)00069-0, 308, 5-6, 375-380, 2003.03, We have studied the string-like structure generated in nanosecond laser-matter interactions. An attempt is made at gaining, from a frozen picture, information on the vortex-filament dynamics by estimating the hydrodynamic parameters of the shear layer of molten metal surface. The parameter values are consistent with the filament instability, and are in a range that admits breakup of waves, formation of loop solitons and reconnections of filaments. (C) 2003 Elsevier Science B.V. All rights reserved.. |

44. |
福本 康秀, 服部 裕司, Curvature instability of a vortex ring (Mathematical Analysis in Fluid and Gas Dynamics), *数理解析研究所講究録*, 1353, 143-162, 2004.10. |

45. |
S Lugomer, Y Fukumoto, Hierarchical instability of a vortex ring array in multipulse laser-matter interactions, *FLUID DYNAMICS RESEARCH*, 10.1016/j.fluiddyn.2004.06.005, 36, 4-6, 277-290, 2005.04, In XeCl excimer Laser interactions with Co-coated steel surfaces, we have seemingly realized a quasi-linear array of vortex rings. These form from instabilities on an array of vortex filaments which emerge and then lead to a series of loops. Finally the collapse-and-reconnection process yields a cascade of nearby vortex rings. Since the filament array is subjected to randomly distributed local multipolar strains, unstable waves can develop on the vortex rings. These deformed shapes are frozen permanently by ultrafast cooling, following the last laser pulse. Using a modal analysis, an attempt is made to relate the wave structures to a parametric resonance instability which is caused by dipolar and a quadrupolar fields. Multipulse laser-matter interactions are thus capable of nonselective excitation of vortex ring instabilities of various sizes and various modal structures. (c) 2005 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.. |

46. |
Y Fukumoto, J Hattori, Curvature instability of a vortex ring, *JOURNAL OF FLUID MECHANICS*, 10.1017/S0022112004002678, 526, 77-115, 2005.03, A global stability analysis of Kelvin's vortex ring to three-dimensional disturbances of infinitesimal amplitude is made. The basic state is a steady asymptotic solution of the Euler equations, in powers of the ratio epsilon of the core radius to the ring radius, for an axisymmetric vortex ring with vorticity proportional to the distance from the symmetric axis. The effect of ring curvature appears at first order, in the form of a dipole field, and a local straining field, which is a quadrupole field, follows at second order. The eigenvalue problem of the Euler equations, retaining the terms to first order, is solved in closed form, in terms of the Bessel and the modified Bessel functions. We show that the dipole field causes a parametric resonance instability between a pair of Kelvin waves whose azimuthal wavenumbers are separated by 1. The most unstable mode occurs in the short-wavelength limit, under the constraint that the radial and the azimuthal wavenumbers are of the same magnitude, and the limiting value of maximum growth rate coincides with the value 165/256 epsilon obtained by Hattori A Fukumoto (Phys. Fluids, vol. 15, 2003, p. 3151) by means of the geometric optics method. The instability mechanism is traced to stretching of disturbance vorticity in the toroidal direction. In the absence of viscosity, the dipole effect outweighs the straining field effect of O(epsilon(2)) known as the Moore-Saffman-Tsai-Widnall instability. The viscosity acts to damp the former preferentially and these effects compete with each other.. |

47. |
4. Vortex patch(Fundamentals of Vortex Motion). |

48. |
3. Vortex sheet(Fundamentals of Vortex Motion). |

49. |
2. Point Vortices(Fundamentals of Vortex Motion). |

50. |
1. Kinematics and Dynamics of Vorticity(Fundamentals of Vortex Motion). |

51. |
7. Three-dimensional motion of vortex tubes(Fundamentals of Vortex Motion). |

52. |
6. Vortex Ring(Fundamentals of Vortex Motion). |

53. |
廣田 真, 福本 康秀, Spectrum and Energy of Waves in Mean Shear Flows (Mathematical Analysis in Fluid and Gas Dynamics), *数理解析研究所講究録*, 1592, 24-40, 2008.04. |

54. |
福本 康秀, Kinematic variational principle for vortical structure of Euler flows and beyond (Modern approach and developments to Onsager's theory on statistical vortices : RIMS合宿型セミナー報告集), *数理解析研究所講究録*, 1798, 198-212, 2012.06. |

55. |
Yasuhide Fukumoto, Rong Zou, Local stability analysis of azimuthal magnetorotational instability of ideal MHD flows, *Progress of Theoretical and Experimental Physics (PTEP)*, 10.1093/ptep/ptu139, 2014, 113J01 (18 pages), 2014.11. |

56. |
Oleg N Kirillov, Frank Stefani, Yasuhide Fukumoto, Local instabilities in magnetized rotational flows: A short-wavelength approach, *Journal of Fluid Mechanics*, 10.1017/jfm.2014.614, 760, 591-633, 2014.11. |

57. |
Youhei Kawazura, Zensho Yoshida, Yasuhide Fukumoto, Relabeling symmetry in relativistic fluids and plasmas, *Journal of Physics A: Mathematical and Theoretical*, 10.1088/1751-8113/47/46/465501, 47, 465501 (17 pages), 2014.10. |

58. |
Jishan Fan, Yasuhide Fukumoto, Yong Zhou, Regularity criteria for the incompressible Hall-MHD system, *Z. Angew. Math. Mech. (ZAMM)*, 10.1002/zamm.201400102, 5 pages, 2014.09. |

59. |
Oleg N. Kirillov, Frank Stefani, Yasuhide Fukumoto, Instabilities of rotational flows in azimuthal magnetic fields of arbitrary radial dependence, *Fluid Dynamics Research*, doi:10.1088/0169-5983/46/3/031403, 2014.06, Using the Wentzel–Kramers–Brillouin (WKB) approximation we perform a linear stability analysis for a rotational flow of a viscous and electrically conducting fluid in an external azimuthal magnetic field that has an arbitrary radial profile $B_{\phi}(R)$. In the inductionless approximation, we find the growth rate of the three-dimensional perturbation in a closed form and demonstrate in particular that it can be positive when the velocity profile is Keplerian and the magnetic field profile is slightly shallower than $R^{-1}$.. |

60. |
Oleg N. Kirillov, Frank Stefani, Yasuhide Fukumoto, Instabilities of rotational flows in azimuthal magnetic fields of arbitrary radial dependence, *Fluid Dynamics Research*, 10.1088/0169-5983/46/3/031403, 46, 3, 2014.06, Using the Wentzel-Kramers-Brillouin (WKB) approximation we perform a linear stability analysis for a rotational flow of a viscous and electrically conducting fluid in an external azimuthal magnetic field that has an arbitrary radial profile B_{φ} (R). In the inductionless approximation, we find the growth rate of the three-dimensional perturbation in a closed form and demonstrate in particular that it can be positive when the velocity profile is Keplerian and the magnetic field profile is slightly shallower than R ^{-1}.. |

61. |
Yuji Hattori, Yasuhide Fukumoto, Modal stability analysis of a helical vortex tube with axial flow, *Journal of Fluid Mechanics*, 10.1017/jfm.2013.591, 738, 222-249, 2014.01, The linear stability of a helical vortex tube with axial flow, which is a model of helical vortices emanating from rotating wings, is studied by modal stability analysis. At the leading order the base flow is set to the Rankine vortex with uniform velocity along the helical tube whose centreline is a helix of constant curvature and torsion. The helical vortex tube in an infinite domain, in which the free boundary condition is imposed at the surface of the tube, is our major target although the case of the rigid boundary condition is also considered in order to elucidate the effects of torsion and the combined effects of torsion and axial flow. The analysis is based on the linearized incompressible Euler equations expanded in ∈ which is the ratio of the core to curvature radius of the tube. The unstable growth rate can be evaluated using the leading-order neutral modes called the Kelvin waves with the expanded equations. At O(∈) the instability is a linear combination of the curvature instability due to the curvature of the tube and the precessional instability due to the axial flow, both parametric instabilities appearing at the same resonance condition. At the next order O(∈^{2}) not only the effects of torsion but also the combined effects of torsion and axial flow appear, a fact which has been shown only for the short-wave limit. The maximum growth rate increases for the right-handed/left-handed helix with positive/negative helicity, in which the torsion makes the period of particle motion increase. All results converge to the previous local stability results in the short-wave limit. The differences between the two cases of different boundary conditions are due to the isolated mode of the free boundary case, whose dispersion curve depends strongly on the axial flow.. |

62. |
Yuji Hattori, Yasuhide Fukumoto, Modal stability analysis of a helical vortex tube with axial flow, *Journal of Fluid Mechanics*, http://dx.doi.org/10.1017/jfm.2013.591, 738, 222-249, 2014.01, The linear stability of a helical vortex tube with axial flow, which is a model of helical vortices emanating from rotating wings, is studied by modal stability analysis. At the leading order the base flow is set to the Rankine vortex with uniform velocity along the helical tube whose centreline is a helix of constant curvature and torsion. The helical vortex tube in an infinite domain, in which the free boundary condition is imposed at the surface of the tube, is our major target although the case of the rigid boundary condition is also considered in order to elucidate the effects of torsion and the combined effects of torsion and axial flow. The analysis is based on the linearized incompressible Euler equations expanded in $\epsilon$ which is the ratio of the core to curvature radius of the tube. The unstable growth rate can be evaluated using the leading-order neutral modes called the Kelvin waves with the expanded equations. At $O(\epsilon)$ the instability is a linear combination of the curvature instability due to the curvature of the tube and the precessional instability due to the axial flow, both parametric instabilities appearing at the same resonance condition. At the next order $O(\epsilon^2)$ not only the effects of torsion but also the combined effects of torsion and axial flow appear, a fact which has been shown only for the short-wave limit. The maximum growth rate increases for the right-handed/left-handed helix with positive/negative helicity, in which the torsion makes the period of particle motion increase. All results converge to the previous local stability results in the short-wave limit. The differences between the two cases of different boundary conditions are due to the isolated mode of the free boundary case, whose dispersion curve depends strongly on the axial flow. . |

63. |
Yasuhide Fukumoto, Makoto Hirota, Youichi Mie, Note on representation of wave energy of a rotating flow in terms of dispersion relation, *Proc. of BIRS Workshop on Spectral Analysis, Stability and Bifurcations in Nonlinear Physical Systems (eds. O. N. Kirillov and D. N. Pelinovsky, Wiley-ISTE, 2014)*, 139-153, 2014.01, A steady Euler flow of an inviscid incompressible fluid is characterized as an extremum of the total kinetic energy with respect to perturbations constrained to an isovortical sheet. We exploit the criticality in the Hamiltonian to calculate the energy of three-dimensional waves on a steady vortical flow, and, as a by-product, to calculate the mean flow, induced by nonlinear interaction of waves with themselves. We show that the energy of waves on a rotating flow is expressible in terms of a derivative of the dispersion relation with respect to the frequency.. |

64. |
Yasuhide Fukumoto, Hamiltonian bifurcation theory for a rotating flow subject to elliptic straining field, 10.1088/0031-8949/2013/T155/014042, 155, 014042-1-014042-10, 2013.12, A weakly nonlinear stability theory is developed for a rotating flow confined in a cylinder of elliptic cross-section. The straining field associated with elliptic deformation of the cross-section breaks the SO(2)-symmetry of the basic flow and amplifies a pair of Kelvin waves whose azimuthal wavenumbers are separated by 2, being referred to as the Moore-Saffman-Tsai-Widnall (MSTW) instability. The Eulerian approach is unable to fully determine the mean flow induced by nonlinear interaction of the Kelvin waves. We establish a general framework for deriving the mean flow by a restriction to isovortical disturbances with use of the Lagrangian variables and put it on the ground of the generalized Lagrangian-mean theory. The resulting formula reveals enhancement of mass transport in regions dominated by the vorticity of the basic flow. With the mean flow at hand, we derive unambiguously the weakly nonlinear amplitude equations to third order for a nonstationary mode. By an appropriate normalization of the amplitude, the resulting equations are made Hamiltonian systems of four degrees of freedom, possibly with three first integrals identifiable as the wave energy and the mean flow.. |

65. |
Jishan Fan, Yasuhide Fukumoto, Yong Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, 10.3934/krm.2013.6.545, 6, 3, 545-556, 2013.09, In this paper, logarithmically improved regularity criteria for the generalized Navier-Stokes equations are established in terms of the velocity, vorticity and pressure, respectively. Here BMO, the Triebel-Lizorkin and Besov spaces are used, which extend usual Sobolev spaces much. Similar results for the quasi-geostrophic ows and the generalized MHD equations are also listed.. |

66. |
Yasuhide Fukumoto, Hirofumi Sakuma, A unified view of topological invariants of barotropic and baroclinic fluids and their application to formal stability analysis of three-dimensional ideal gas flows, 10.1016/j.piutam.2013.03.025, 7, 213-222, 2013.04, Noether's theorem associated with the particle relabeling symmetry group leads us to a unified view that all the topological invariants of a barotropic fluid are variants of the cross helicity. The similar is shown to be true of a baroclinic fluid. A cross-helicity representation is given to the Casimir invariant, a class of integrals including an arbitrary function of the specific entropy and the potential vorticity. We then develop a new energy-Casimir convexity method for three-dimensional stability of equilibria of general rotating flows of an ideal baroclinic gas, without appealing to the Boussinesq approximation. By fully exploiting the Casimir invariant, we have succeeded in ruling out a term including the gradient of a dependent variable from the energy-Casimir function and have established a sharp linear stability criterion, being an extension of the Richardson-number criterion.. |

67. |
Yasuhide Fukumoto, A. B. Samokhin, Singular electromagnetic modes in an anisotropic medium, 10.1016/j.wavemoti.2012.11.001, 50, 481-493, 2013.03, We construct the singular mode corresponding to a spatial essential spectrum of the integral operator for the scattering of the electromagnetic waves by a three-dimensional body of finite size with inhomogeneous and anisotropic dielectric permittivity tensor. The permittivity tensor field is assumed to be H\"older continuous throughout the whole space. The singular volume integral equation, transformed from Maxwell's equations, makes it feasible to deduce explicit form of both the continuous essential spectrum and the corresponding singular modes. The obtained singular mode is a natural extension of the previously obtained one for the isotropic case and is applicable to a much wider class of dielectric scattering bodies. A discussion is made of possibility for realizability of the electromagnetic waves, with finite energy, concentrated at a point in the body.. |

68. |
Abuduwaili Paerhati, Yasuhide Fukumoto, An example exempted from Thomson-Tait-Chetayev's theorem, 10.7566/JPSJ.82.043002, 82, 3, 043002-1-043002-4, 2013.03, An example is given of a mechanical system whose behavior does not follow Thomson-Tait-Chetayev's (TTC) theorem which states that, for a system with an unstable potential, a state stabilized by gyroscopic forces goes unstable after the addition of arbitrary dissipation. The example is brought by a system, closely related with a heavy symmetrical top, describing motion of a charged spherical pendulum subjected to the Lorentz force, in a magnetic-monopole field sitting at the sphere center, as well as the gravity force. A drag force proportional to the velocity is exerted on the pendulum. The upright state, an equilibrium stabilized by the Lorentz force, is shown to be exempted from the TTC theorem. The numerical calculation of the full nonlinear system is performed for precession. For the slow precession, the drag force acts to continuously tilt down the top axis toward vertically downward equilibrium, following the dissipation-induced instability. On the contrary, for the fast precession, the drag acts to continuously tilt up the axis against the gravity force, despite losing the energy.. |

69. |
Oleg N. Kirillov, Frank Stefani, Yasuhide Fukumoto, A unifying picture of helical and azimuthal MRI, and the universal significance of the Liu limit, 10.1088/0004-637X/756/1/83, 756, 1, 756:83-1-756:83-6, 2012.09, The magnetorotational instability (MRI) plays a key role for cosmic structure formation by triggering turbulence in the rotating flows of accretion disks that would be otherwise hydrodynamically stable. In the limit of smallmagnetic Prandtl number, the helical and the azimuthal versions of MRI are known to be governed by a quite different scaling behavior than the standard MRI with a vertical app ied magnetic field. Using the short-wavelength approximation for an incompressible, resistive, and viscous rotating fluid, we present a unified description of helical and azimuthal MRI, and we identify the universal character of the Liu limit 2(1−2^{1/2})= 0.8284 for the critical Rossby number. From this universal behavior we are also led to the prediction that the instability will be governed by a mode with an azimuthal wavenumber that is proportional to the ratio of axial to azimuthal applied magnetic field, when this ratio becomes large and the Rossby number is close to the Liu limit.. |

70. |
Y. Hattori, Y. Fukumoto, Effects of axial flow on the stability of a helical vortex tube, *Physics of Fluids*, 10.1063/1.4717769, 24 , 5, 054102 (15 pages) , 2012.05, The effects of axial flow on the stability of a helical vortex tube are studied by short-wavelength stability analysis. By axial flow we mean the flow along the helical tube inside the vortex core. At the leading order the base flow is set to the Rankine vortex with uniform velocity along the helical tube. The exponential growth rate is obtained analytically as the magnitude of he sum of three O(1) and five O(2) complex numbers, where ϵ is the ratio of the core to curvature radius. At O(1) the effect of axial flow can be regarded as the effect of the Coriolis force; as a result the instability is the superposition of the curvature instability and the Coriolis or precessional instability since the two instabilities occur under the same resonance condition. At O(2) combined effects of the axial flow and the torsion appear; the maximum growth rate increases when the period of particle motion increases.. |

71. |
F. Kaplanski, Y. Fukumoto, Y. Rudi, Reynolds-number effect on vortex ring evolution in a viscous fluid, *Physics of Fluids*, 10.1063/1.3693276, 24, 3, 033101 (13 pages) , 2012.03, An analytical model describing vortex ring for low Reynolds numbers proposed previously by Kaplanski and Rudi [Phys. Fluids,17, 087101 (2005)], is extended to a vortex rings for high Reynolds numbers. The experimental results show that the vortex ring core takes the oblate ellipsoidal shape with increasing $Re$. In order to model this feature, we suggest an expression for the vorticity distribution, which corrects the linearized solution of the Navier-Stokes equation, with two disposable nondimensional parameters $\lambda$ and $\beta$ governing the shape of the vortex core, and derive the new expressions for the translation velocity, energy, circulation and streamfunction on the basis of it. The appropriate values of $\lambda$ and $\beta$ are calculated by equating the nondimensional energy $E_d$ and circulation $\Gamma_d$ of the theoretical vortex to the corresponding values obtained from the experimental or numerical vortex ring. To validate the model, the data adapted from the numerical study of vortex ring at $Re=1400$ performed by Danaila and Helie [Phys. Fluids,20, 073602 (2008)], is applied. It is shown that the temporal evolution of the translation velocity at high Reynolds numbers based on these data compares well with the experiments and numerical simulations.. |

72. |
Y. Fukumoto, M. Hirota, Y. Mie, Energy and mean flow of Kelvin waves, and their application to weakly nonlinear stability of an elliptical flow, *Proc. of the International Conference ‘Mathematical Analysis on the Navier-Stokes Equations and Related Topics, Past and Future ―in memory of Professor Tetsuro Miyakawa', Mathematical Sciences and Applications *, 43, 53-70, 2011.12, We establish a Lagrangian method which provides us with an unambiguous definition of the wave energy and facilitates its calculation. A steady incompressible Euler flow is characterized as a state of the maximum of the total kinetic energy with respect to disturbances constrained to an isovortical sheet, and the isovortical disturbances are tractable only in terms of the Lagrangian variables. The criticality in energy of a steady flow allows us to work out the wave energy of Kelvin waves solely from the linear disturbance field. As a by-product, the mean flow of second order in amplitude, induced by the nonlinear interaction of Kelvin waves, is obtainable, which provides us with a bypass to enter into weakly nonlinear regime of amplitude evolution.. |

73. |
Me Me Naing, Y. Fukumoto, Local instability of a rotating flow driven by precession of arbitrary frequency, *Fluid Dynamics Research*, 10.1088/0169-5983/43/5/055502, 43, 055502 (11 pages) , 2011.08, We revisit the local stability, to three-dimensional disturbances, of rotating flows with circular streamlines, whose rotation axis executes constant precessional motion about an axis perpendicular to itself. In the rotating frame, the basic flow is steady velocity field linear in coordinates in an unbounded domain constructed by Kerswell (1993), and admits the use of the WKB method. For small precession frequency, we recover Kerswell's result. A novel instability is found at large frequency for which the axial wavenumber executes an oscillation around zero; drastic growth of disturbance amplitude occurs only in an extremely short time interval around the time where the axial wavenumber vanishes. In the limit of infinite precession frequency, the growth rate exhibits singular behavior with respect to a parameter characterizing the tilting angle of the wave vector.. |

74. |
Y. Fukumoto, M. Hirota, Y. Mie, Lagrangian approach to weakly nonlinear stability of elliptical flow, *Physica Scripta T*, 10.1088/0031-8949/2010/T142/011003, 142, 014049 (7 pages), 2010.03. |

75. |
Y. Mie, Y. Fukumoto, Weakly nonlinear saturation of stationary resonance of a rotating flow in an elliptic cylinder, *Journal of Math-for-Industry*, 2, A, 27-37, 2010.04. |

76. |
F. Kaplanski, S. S. Sazhin, S. Begg, Y. Fukumoto, M. Heikal, Dynamics of vortex rings and spray induced vortex ring-like structures, *European J. Mechanics B/ Fluids*, 29, 208-216, 2010.03. |

77. |
S. Lugomer, Y. Fukumoto, Generation of ribbons, helicoids and complex Scherk surface in laser-matter interactions, *Physical Reviews E*, 81, 036311 (11 pages), 2010.03. |

78. |
Y. Hattori, Y. Fukumoto, Short-wave stability of a helical vortex tube: the effect of torsion on the curvature instability, *Theoretical and Computational Fluid Dynamics*, 24, 1-4, 363-368, 2010.03. |

79. |
Y. Fukumoto, Global time evolution of viscous vortex rings, *Theoretical and Computational Fluid Dynamics*, 24, 1-4, 335-347, 2010.03. |

80. |
Me Me Naing, Y. Fukumoto, Local instability of an elliptical flow subjected to a Coriolis force, *J. Phys. Soc. Japan*, 78, 12, 124401 (7 pages), 2009.12. |

81. |
F. Kaplanski, S. Sazhin, Yasuhide Fukumoto, B. Steven, H. Morgan, A generalised vortex ring model, *Journal of Fluid Mechanics*, Vol.622, pp.233-258, 2009.03. |

82. |
Yuji Hattori, Yasuhide Fukumoto, Short-wavelength stability analysis of a helical vortex tube, *Physics of Fluids*, Vol.21, No.1, 014104 (7 pages), 2009.01. |

83. |
Yasuhide Fukumoto, A unified view of topological invariants of fluid flows, *Topologica*, Vol.1, 003 (12 pages), 2008.12. |

84. |
M. Hirota , Y. Fukumoto, Action-angle variables for the continuous spectrum of ideal magnetohydrodynamics, *Physics of Plasmas*, Vol.15, 122101 (11 pages), 2008.12. |

85. |
Yasuhide Fukumoto, M. Hirota, Elliptical instability of a vortex tube and drift current induced by it, *Physica Scripta*, Vol.T132, 014041 (9 pages), 2008.10. |

86. |
M. Hirota, Y. Fukumoto, Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra, *Journal of Mathematical Physics*, Vol.49, 083101 (28 pages), 2008.10. |

87. |
Yasuhide Fukumoto, H. K. Moffatt, Kinematic variational principle for motion of vortex rings, *Physica D*, Vol.237, No.14-17, pp.2210-2217, 2008.08. |

88. |
Yasuhide Fukumoto, Analogy of a vortex-jet filament with the Kirchhoff elastic rod and its dynamical extension, *Proc. of IUTAM Symposium on on Hamiltonian dynamics, vortex structure and turbulence, IUTAM Bookseries Vol. 6 (eds. A.V. Borisov, V.V. Kozlov, I.S. Manaev and M.A. Sokolovskiy, Springer, 2008)*, IUTAM Bookseries Vol. 6，pp. 77-87, 2008.03. |

89. |
S. Lugomer, Yasuhide Fukumoto, B. Farkas, T. Szorenyi, A. Toth, Super-complex wave-vortex multiscale phenomena induced in laser-matter interactions, *Physical Reviews E*, Vol. 39, pp. 016305-1-15, 2007.07. |

90. |
Yasuhide Fukumoto, Analogy between a vortex-jet filament and the Kirchhoff elastic rod, *Fluid Dynamics Research*, Vol. 39, No.7 , pp. 511-520., 2007.07. |

91. |
Yasuhide Fukumoto, V. L. Okulov, The velocity field induced by a helical vortex tube, *Physics of Fluids*, 10.1063/1.2061427, 17, 10, Vol. 17, No.10, pp. 107101-1-19, 2005.10. |

92. |
Y.Fukumoto and S. Lugomer, ierarchical instability of a vortex ring array in multipulse laser-matter interactions, *Fluid Dyn. Res.*, in press, 2005.01. |

93. |
Y.Fukumoto and Y.hattori, Curvature instability of a vortex ring, *J. Fluid Mech.*, 10.1017/S0022112004002678, 526, 77-115, 526, 77-115, 2005.01. |

94. |
Y.Fukumoto and V. L. Okulov, Helical dipole, *Doklady Physics*, 10.1134/1.1831533, 49, 11, 662-667, 49, 662-667, 2004.01. |

95. |
Y.Fukumoto, Conservation laws of circulation and helicity as Noether's theorem, *Comput. Fluid Dyn. J.*, 13, 417--421, 2004.01. |

96. |
Yasuhide Fukumoto, The three-dimensional instability of a strained vortex tube revisited, *Journal of Fluid Mechanics*, 10.1017/S0022112003006025, 493, 287-318, Vol.493, pp.287-318, 2003.10. |

97. |
Yuji Hattori, Yasuhide Fukumoto, Short-wavelength stability analysis of thin vortex rings, *Physics of Fluids*, 10.1063/1.1606446, 15, 10, 3151-3163, Vol.15, No.10, pp.3151-3163, 2003.10. |

98. |
Yasuhide Fukumoto, S. Lugomer, Instability of vortex filaments in laser-matter interactions, *Physics Letters A*, 10.1016/S0375-9601(03)00069-0, 308, 5-6, 375-380, Vol.308, No.15,6, pp.375-380, 2003.03. |

99. |
Yasuhide Fukumoto, Yuji Hattori, Linear stability of a vortex ring revisited, *Proceedings of IUTAM Symposium on Tubes, Sheets and Singuraities in Fluid Dynamics (eds. H. K. Moffatt and K. Bajer, Kluwer)*, Fluid Mechanics and Applications series, Vol.71, pp.37-48, 2002.01. |

100. |
Yasuhide Fukumoto, Three-dimensional motion of a vortex filament and its relation to the localized induction hierarchy, *European Physical Journal B*, 10.1140/epjb/e2002-00279-5, 29, 2, 167-171, Vol.29, No.2, pp.167-171, 2002.09. |

101. |
Yasuhide Fukumoto, Higher-order asymptotic theory for the velocity field induced by an inviscid vortex ring, *Fluid Dynamics Research*, 10.1016/S0169-5983(01)00044-2, 30, 2, 65-92, Vol.30, No.2, pp.65-92, 2002.02. |

102. |
Yasuhide Fukumoto, H. K. Moffatt, Motion and expansion of a viscous vortex ring: Elliptical slowing down and diffusive expansion, *Proceedings of Symposium on Turbulence Structure and Vortex Dynamics, Isaac Newton Institute Series (eds. J. C. R. Hunt and J. C. Vassilicos, Cambridge University Press)*, 1-22, pp.1-22, 2000.01. |

103. |
Yasuhide Fukumoto, Motion of a curved vortex filament: Higher-order asymptotics, *Proceedings of IUTAM Symposium on Geometry and Statistics of Turbulence (eds. T. Kambe, T. Nakano and T. Miyauchi, Kluwer)*, 59, 211-216, Fluid Mechanics and Applications series, Vol.59, pp.211-216, 2001.01. |

104. |
福本 康秀, Biot-Savart 則に対する Dyson の方法再考, *ながれ*, Vol.19, No.3, pp.180-185, 2000.06. |

105. |
T. Rozi, Yasuhide Fukumoto, Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity, *Journal of the Physical Society of Japan*, Vol.69, No.8, pp.2700-2701, 2000.08. |

106. |
Yasuhide Fukumoto, H. K. Moffatt, Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity, *Journal of Fluid Mechanics*, 10.1017/S0022112000008995, 417, 1-45, Vol.417, pp.1-45, 2001.08. |

107. |
T. Rozi, Yasuhide Fukumoto, The response of Hill's vortex to a small three dimensional disturbance in the case of m=5, *In Progress in Experimental and Computational Mechanics in Engineering and Material Behaviour (eds. D. Zhu, M. Kikuchi, Y. Shen and M. Geni, Northwestern Ploytech. Univ. Press)*, pp.346-351, 1999.09. |

108. |
Yasuhide Fukumoto, H. K. Moffatt, Motion of a vortex ring in a viscous fluid: Higher-order asymptotics, *Proceedings of IUTAM Symposium on Dynamics of Slender Vortices (eds. E. Krause and K. Gersten, Kluwer)*, 44, 21-34, Fluid Mechanics and Applications series, Vol.44, pp.21-34, 1998.01. |

109. |
Yasuhide Fukumoto, Stationary configurations of a vortex filament in background flows, *Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences*, Vol.453, No.1961, pp.1205-1232, 1997.06. |

110. |
Yasuhide Fukumoto, Mitsuharu Miyajima, The localised induction hierarchy and the Lund-Regge equation, *Journal of Physics A - Mathmatical and General*, 10.1088/0305-4470/29/24/025, 29, 24, 8025-8034, Vol.29, No.24, pp.8025-8034, 1996.12. |

111. |
T MIYAZAKI, T IMAI, Y FUKUMOTO, 3-DIMENSIONAL INSTABILITY OF KIRCHHOFF ELLIPTIC VORTEX, *PHYSICS OF FLUIDS*, 10.1063/1.868719, 7, 1, 195-202, 1995.01. |