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Lyapunov stability of ideal compressible and incompressible fluid equilibria in three dimensions

Description: Linearized stability of ideal compressible and incompressible fluid equilibria in three dimensions is analyzed using Lyapunov's direct method. An action principle is given for the Eulerian and Lagrangian fluid descriptions and the family of constants of motion due to symmetry under fluid-particle relabelling is derived in the form of Ertel's theorem for each description. In an augmented Euleriah description, the steady equilibrium flows of these two fluids theories are identified as critical points of the conserved Lyapunov functionals defined by the sum, H + C, of the energy H, and the Ertel constants of motion, C. It turns out that unconditional linear Lyapunov stability of these flows in the norm provided by the second variation of H + C is precluded by vortex-particle stretching, even for otherwise shear-stable flows. Conditional Lyapunov stability of these flows is discussed. 24 refs.
Date: August 1, 1985
Creator: Holm, D.D.
Partner: UNT Libraries Government Documents Department

Stability of planar multifluid plasma equilibria by Arnold's method

Description: A method developed by Arnold to prove nonlinear stability of certain steady states for ideal incompressible flow in two dimensions is extended to the case of barotropic, compressible, multifluid plasmas. This extension is accomplished by constructing conserved functionals derived from degeneracy of Poisson brackets. The results are applied to planar shear flows of the plasma.
Date: January 1, 1983
Creator: Holm, D.D.
Partner: UNT Libraries Government Documents Department

Gyroscopic analog for magnetohydrodynamics

Description: The gross features of plasma equilibrium and dynamics in the ideal magnetohydrodynamics (MHD) model can be understood in terms of a dynamical system which closely resembles the equations for a deformable gyroscope.
Date: January 1, 1981
Creator: Holm, D.D.
Partner: UNT Libraries Government Documents Department

Order and chaos in polarized nonlinear optics

Description: Methods for investigating temporal complexity in Hamiltonian systems are applied to the dynamics of a polarized optical laser beam propagating as a travelling wave in a medium with cubically nonlinear polarizability (i.e., a Kerr medium). The theory of Hamiltonian systems with symmetry is used to study the geometry of phase space for the optical problem, transforming from C{sup 2} to S{sup 2} {times} (J,{theta}), where (J,{theta}) is a symplectic action-angle pair. The bifurcations of the phase portraits of the Hamiltonian motion on S{sup 2} are classified and shown graphically. These bifurcations create various saddle connections on S{sup 2} as either J (the beam intensity), or the optical parameters of the medium are varied. After this bifurcation analysis, the Melnikov method is used to demonstrate analytically that the saddle connections break and intersect transversely in a Poincare map under spatially periodic perturbations of the optical parameters of the medium. These transverse intersections in the Poincare map imply intermittent polarization switching with extreme sensitivity to initial conditions characterized by a Smale horseshoe construction for the travelling waves of a polarized optical laser pulse. The resulting chaotic behavior in the form of sensitive dependence on initial conditions may have implications for the control and predictability of nonlinear optical polarization switching in birefringent media. 19 refs., 2 figs., 1 tab.
Date: January 1, 1990
Creator: Holm, D.D.
Partner: UNT Libraries Government Documents Department

Spherical shock collapse in a non-ideal medium

Description: Non-ideal fluid motions are studied. Spherical self-similar convergence is calculated for a strong shock in an ideal medium. Group theory is used to place a symmetry condition on the adiabatic bulk modulus, B/sub s/(p,v), for which three independent scale transformations of Euler's equations are admitted. The types of non-ideal media which satisfy the bulk modulus symmetry condition include equations of state of Mie-Grusneisen type. Thus the theory applies to a wide class of materials. In particular it applies to non-degenerate solids at shock pressures well above the yield stress. (GHT)
Date: January 1, 1978
Creator: Axford, R.A. & Holm, D.D.
Partner: UNT Libraries Government Documents Department

Dispersive water waves in one and two dimensions

Description: This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). We derived and analyzed new shallow water equations for one-dimensional flows near the critical Froude number as well as related integrable systems of evolutionary nonlinear partial differential equations in one spatial dimension, while developing new directions for the mathematics underlying the integrability of these systems. In particular, we applied the spectrum generating equation method to create and study new integrable systems of nonlinear partial differential equations related to our integrable shallow water equations. We also investigated the solutions of these systems of equations on a periodic spatial domain by using methods from the complex algebraic geometry of Riemann surfaces. We developed certain aspects of the required mathematical tools in the course of this investigation, such as inverse scattering with degenerate potentials, asymptotic reduction of the angle representations, geometric singular perturbation theory, modulation theory and singularity tracking for completely integrable equations. We also studied equations that admit weak solutions, i.e., solutions with discontinuous derivatives in the form of comers or cusps, even though they are solutions of integrable models, a property that is often incorrectly assumed to imply smooth solution behavior. In related work, we derived new shallow water equations in two dimensions for an incompressible fluid with a free surface that is moving under the force of gravity. These equations provide an estimate of the long-time asymptotic effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity, and they describe the flow regime in which the Froude number is small -- much smaller even than the small aspect ratio of the shallow domain.
Date: August 1, 1997
Creator: Holm, D.D. & Camassa, R.A.
Partner: UNT Libraries Government Documents Department

Proceedings of the conference on numerical methods in high temperature physics

Description: These proceedings contain full papers presented at the Los Alamos Conference on High Temperature Physics. This conference discussed many aspects of high temperature physics including hydrodynamics, radiation and particle transport and some computational issues important for efficient calculations. The meetings was held between researchers from Los Alamos and the French Commissariat a L'Energy Atomique (CEA).
Date: November 1, 1988
Creator: Alcouffe, R.E.; Holm, D.D. & O'Rourke, P.J. (comps.)
Partner: UNT Libraries Government Documents Department

Finite dimensionality in the complex Ginzburg-Landau equation

Description: Finite dimensionality is shown to exist in the complex Ginzburg-Landau equation periodic on the interval (0,1). A cone condition is derived and explained which gives upper bounds on the number of Fourier modes required to span the universal attractor and hence upper bounds on the attractor dimension itself. In terms of the parameter R these bounds are not large. For instance, when vertical bar ..mu.. vertical bar less than or equal to ..sqrt..3, the Fourier spanning dimension is 0(R/sup 3/2/). Lower bounds are estimated from the number of unstable side-bands using ideas from work on the Eckhaus instability. Upper bounds on the dimension of the attractor itself are obtained by bounding (or, for vertical bar ..mu.. vertical bar less than or equal to ..sqrt..3, computing exactly) the Lyapunov dimension and invoking a recent theorem of Constantin and Foias, which asserts that the Lyapunov dimension, defined by the Kaplan-Yorke formula, is an upper bound on the Hausdorff dimension. 39 refs., 7 figs.
Date: January 1, 1987
Creator: Doering, C.R.; Gibbon, J.D.; Holm, D.D. & Nicolaenko, B.
Partner: UNT Libraries Government Documents Department

Toward an extended-geostrophic Euler-Poincare model for mesoscale oceanographic flow

Description: The authors consider the motion of a rotating, continuously stratified fluid governed by the hydrostatic primitive equations (PE). An approximate Hamiltonian (L1) model for small Rossby number {var_epsilon} is derived for application to mesoscale oceanographic flow problems. Numerical experiments involving a baroclinically unstable oceanic jet are utilized to assess the accuracy of the L1 model compared to the PE and to other approximate models, such as the quasigeostrophic (QG) and the geostrophic momentum (GM) equations. The results of the numerical experiments for moderate Rossby number flow show that the L1 model gives accurate solutions with errors substantially smaller than QG or GM.
Date: July 1998
Creator: Allen, J. S.; Newberger, P. A. & Holm, D. D.
Partner: UNT Libraries Government Documents Department

Dispersive internal long wave models

Description: This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) project at Los Alamos National Laboratory (LANL). This work is a joint analytical and numerical study of internal dispersive water wave propagation in a stratified two-layer fluid, a problem that has important geophysical fluid dynamics applications. Two-layer models can capture the main density-dependent effects because they can support, unlike homogeneous fluid models, the observed large amplitude internal wave motion at the interface between layers. The authors have derived new model equations using multiscale asymptotics in combination with the method they have developed for vertically averaging velocity and vorticity fields across fluid layers within the original Euler equations. The authors have found new exact conservation laws for layer-mean vorticity that have exact counterparts in the models. With this approach, they have derived a class of equations that retain the full nonlinearity of the original Euler equations while preserving the simplicity of known weakly nonlinear models, thus providing the theoretical foundation for experimental results so far unexplained.
Date: November 1, 1998
Creator: Camassa, R.; Choi, W.; Holm, D.D.; Levermore, C.D. & Lvov, Y.
Partner: UNT Libraries Government Documents Department

Chaotic behavior in nonlinear polarization dynamics

Description: We analyze the problem of two counterpropagating optical laser beams in a slightly nonlinear medium from the point of view of Hamiltonian systems; the one-beam subproblem is also investigated as a special case. We are interested in these systems as integrable dynamical systems which undergo chaotic behavior under various types of perturbations. The phase space for the two-beam problem is C{sup 2} {times} C{sup 2} when we restricted the the regime of travelling-wave solutions. We use the method of reduction for Hamiltonian systems invariant under one-parameter symmetry groups to demonstrate that the phase space reduces to the two-sphere S{sup 2} and is therefore completely integrable. The phase portraits of the system are classified and we also determine the bifurcations that modify these portraits; some new degenerate bifurcations are presented in this context. Finally, we introduce various physically relevant perturbations and use the Melnikov method to prove that horseshoe chaos and Arnold diffusion occur as consequences of these perturbations. 10 refs., 7 figs., 1 tab.
Date: January 1, 1989
Creator: David, D.; Holm, D.D. & Tratnik, M.V. (Los Alamos National Lab., NM (USA))
Partner: UNT Libraries Government Documents Department

Hamiltonian chaos in a nonlinear polarized optical beam

Description: This lecture concerns the applications of ideas about temporal complexity in Hamiltonian systems to the dynamics of an optical laser beam with arbitrary polarization propagating as a traveling wave in a medium with cubically nonlinear polarizability. We use methods from the theory of Hamiltonian systems with symmetry to study the geometry of phase space for this optical problem, transforming from C{sup 2} to S{sup 3} {times} S{sup 1}, first, and then to S{sup 2} {times} (J, {theta}), where (J, {theta}) is a symplectic action-angle pair. The bifurcations of the phase portraits of the Hamiltonian motion on S{sub 2} are classified and displayed graphically. These bifurcations take place when either J (the beam intensity), or the optical parameters of the medium are varied. After this bifurcation analysis has shown the existence of various saddle connections on S{sup 2}, the Melnikov method is used to demonstrate analytically that the traveling-wave dynamics of a polarized optical laser pulse develops chaotic behavior in the form of Smale horseshoes when propagating through spatially periodic perturbations in the optical parameters of the medium. 20 refs., 7 figs.
Date: January 1, 1989
Creator: David, D.; Holm, D.D. & Tratnik, M.V. (Los Alamos National Lab., NM (USA))
Partner: UNT Libraries Government Documents Department

Self-Consistent Multiscale Theory of Internal Wave, Mean-Flow Interactions

Description: This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) project at Los Alamos National Laboratory (LANL). The research reported here produced new effective ways to solve multiscale problems in nonlinear fluid dynamics, such as turbulent flow and global ocean circulation. This was accomplished by first developing new methods for averaging over random or rapidly varying phases in nonlinear systems at multiple scales. We then used these methods to derive new equations for analyzing the mean behavior of fluctuation processes coupled self consistently to nonlinear fluid dynamics. This project extends a technology base relevant to a variety of multiscale problems in fluid dynamics of interest to the Laboratory and applies this technology to those problems. The project's theoretical and mathematical developments also help advance our understanding of the scientific principles underlying the control of complex behavior in fluid dynamical systems with strong spatial and temporal internal variability.
Date: June 3, 1999
Creator: Holm, D.D.; Aceves, A.; Allen, J.S.; Alber, M.; Camassa, R.; Cendra, H. et al.
Partner: UNT Libraries Government Documents Department