Publications
335 results found
Holm DD, Tronci C, 2009, Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 465, Pages: 457-476, ISSN: 1364-5021
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- Citations: 17
Holm DD, Naraigh LO, Tronci C, 2009, Singular solutions of a modified two-component Camassa-Holm equation, Physical Review E, Vol: 79, ISSN: 1539-3755
The Camassa-Holm (CH) equation is a well-known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow a dependence on the average density as well as the pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in the velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. Numerical results for the MCH2 system are given and compared with the pure CH2 case. These numerics show that the modification in the MCH2 system to introduce the average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for the MCH2 system shows a different asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, the MCH2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes.
Kuczaj AK, Geurts BJ, Holm DD, 2009, Intermittency effects in rotating decaying turbulence, Pages: 791-796
Rotation modulates turbulence causing columnar structuring of a turbulent flow in case of sufficiently strong rotation. This yields significant changes in the flow characteristics and dispersion properties, which makes rotational turbulence modulation particularly relevant in the context of atmospheric and oceanic flows. Here we investigate the canonical flow of turbulence in a periodic box, subjected to rotation about a fixed vertical axis. As point of reference we consider direct numerical simulations of homogeneous isotropic turbulence. Modulation due to rotation at various rotation rates (i.e., different Rossby numbers) is investigated. Special attention is paid to the alteration of intermittency, which is measured in terms of changes in the scaling of the structure functions. A reduction of intermittency quantified with the longitudinal structure functions in the direction perpendicular to the rotation axes will be presented. These numerical findings correspond well to recent results obtained in experiments by Seiwert et al. (2008) [1].
Cotter CJ, Holm DD, 2009, Momentum Maps for Lattice EPDiff, SPECIAL VOLUME: COMPUTATIONAL METHODS FOR THE ATMOSPHERE AND THE OCEANS, Vol: 14, Pages: 247-278, ISSN: 1570-8659
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- Citations: 1
Cotter CJ, Holm DD, 2009, Discrete momentum maps for lattice EPDiff, Handbook of Numerical Analysis, Editors: Temam, Tribbia, Publisher: North-Holland, Pages: 247-278, ISBN: 978-0-444-51893-4
Cotter CJ, Holm DD, 2009, Continuous and discrete Clebsch variational principles, Foundations of Computational Mathematics, Vol: 9
Brody DC, Ellis DCP, Holm DD, 2008, Hamiltonian statistical mechanics, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, Vol: 41, ISSN: 1751-8113
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- Citations: 7
Holm DD, Putkaradze V, Tronci C, 2008, Geometric gradient-flow dynamics with singular solutions, PHYSICA D-NONLINEAR PHENOMENA, Vol: 237, Pages: 2952-2965, ISSN: 0167-2789
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- Citations: 9
Holm DD, Putkaradze V, Tronci C, 2008, Kinetic models of oriented self-assembly, Meeting held in Honor of Darryl D Holms on Geometry and Analysis in Physical Systems, Publisher: IOP PUBLISHING LTD, ISSN: 1751-8113
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- Citations: 4
Holm DD, Putkaradze V, Tronci C, 2008, Kinetic models of oriented self-assembly - art. no. 344010, Meeting held in Honor of Darryl D Holms on Geometry and Analysis in Physical Systems, Pages: 44010-44010
New kinetic models of dissipation are proposed for the dynamics of an ensemble of interacting oriented particles, for example, moving magnetized nano-particles. This is achieved by introducing double-bracket dissipation into kinetic equations by using an oriented Poisson bracket and employing the moment method to derive continuum equations for the evolution of magnetization and mass density. These continuum equations generalize the Debye-Huckel equations for attracting round particles, and Landau-Lifshitz Gilbert equations for spin waves in magnetized media. The dynamics of self-assembly is investigated as the emergent concentration into singular clumps of aligned particles (orientons) starting from random initial conditions. Finally, the theory is extended to describe the dissipative motion of self-interacting curved filaments.
Hecht MW, Holm DD, Petersen MR, et al., 2008, The LANS-α and Leray turbulence parameterizations in primitive equation ocean modeling, Meeting held in Honor of Darryl D Holms on Geometry and Analysis in Physical Systems, Publisher: IOP Publishing Ltd, ISSN: 1751-8113
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- Citations: 16
Percival JR, Cotter CJ, Holm DD, 2008, A Euler–Poincaré framework for the multilayer Green–Nagdhi equations, Meeting held in Honor of Darryl D Holms on Geometry and Analysis in Physical Systems, Publisher: IOP Publishing, Pages: 344018-344031, ISSN: 1751-8113
The Green–Nagdhi equations are frequently used as a model of the wave-like behaviour of the free surface of a fluid, or the interface between two homogeneous fluids of differing densities. Here we show that their multilayer extension arises naturally from a framework based on the Euler–Poincaré theory under an ansatz of columnar motion. The framework also extends to the travelling wave solutions of the equations. We present numerical solutions of the travelling wave problem in a number of flow regimes. We find that the free surface and multilayer waves can exhibit intriguing differences compared to the results of single layer or rigid lid models.
Gibbons J, Holm DD, Tronci C, 2008, Geometry of Vlasov kinetic moments: A bosonic Fock space for the symmetric Schouten bracket, PHYSICS LETTERS A, Vol: 372, Pages: 4184-4196, ISSN: 0375-9601
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- Citations: 19
Hecht MW, Holm DD, Petersen MR, et al., 2008, Implementation of the LANS-α turbulence model in a primitive equation ocean model, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 227, Pages: 5691-5716, ISSN: 0021-9991
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- Citations: 30
Holm DD, Naraigh LO, Tronci C, 2008, Emergent singular solutions of nonlocal density-magnetization equations in one dimension, Physical Review E, Vol: 77, ISSN: 1539-3755
We investigate the emergence of singular solutions in a nonlocal model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the nonlocal effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of nonlocality on the system’s stability properties. We carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data. The singular solutions represent a collection of interacting particles (clumpons). By restricting ourselves to the two-clumpon case, we are reduced to a two-dimensional dynamical system that is readily analyzed, and thus we classify the different clumpon interactions possible.
Graham JP, Holm DD, Mininni PD, et al., 2008, Three regularization models of the Navier-Stokes equations, Physics of Fluids, Vol: 20, ISSN: 1070-6631
We determine how the differences in the treatment of the subfilter-scale physics affect the properties of the flow for three closely related regularizations of Navier–Stokes. The consequences on the applicability of the regularizations as subgrid-scale (SGS) models are also shown by examining their effects on superfilter-scale properties. Numerical solutions of the Clark-α model are compared to two previously employed regularizations, the Lagrangian-averaged Navier–Stokes α-model (LANS-α) and Leray-α, albeit at significantly higher Reynolds number than previous studies, namely, Re≈3300, Taylor Reynolds number of Reλ≈790, and to a direct numerical simulation (DNS) of the Navier–Stokes equations. We derive the de Kármán–Howarth equation for both the Clark-α and Leray-α models. We confirm one of two possible scalings resulting from this equation for Clark-α as well as its associated k−1 energy spectrum. At subfilter scales, Clark-α possesses similar total dissipation and characteristic time to reach a statistical turbulent steady state as Navier–Stokes, but exhibits greater intermittency. As a SGS model, Clark-α reproduces the large-scale energy spectrum and intermittency properties of the DNS. For the Leray-α model, increasing the filter width α decreases the nonlinearity and, hence, the effective Reynolds number is substantially decreased. Therefore, even for the smallest value of α studied Leray-α was inadequate as a SGS model. The LANS-α energy spectrum ∼k1, consistent with its so-called “rigid bodies,” precludes a reproduction of the large-scale energy spectrum of the DNS at high Re while achieving a large reduction in numerical resolution. We find, however, that this same feature reduces its intermittency compared to Clark-α (which shares a similar de Kármán–Howarth equat
Holm DD, 2008, Geometric mechanics, Part I: Dynamics and symmetry, ISBN: 9781848161955
This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduate and beginning graduate students in mathematics, physics and engineering. It treats the dynamics of ray optics, resonant oscillators and the elastic spherical pendulum from a unified geometric viewpoint, by formulating their solutions using reduction by Lie-group symmetries. The only prerequisites are linear algebra, calculus and some familiarity with the Euler–Lagrange variational principles and canonical Poisson brackets in classical mechanics at the beginning undergraduate level. The ideas and concepts of geometric mechanics are explained in the context of explicit examples. Through these examples, the student develops skills in performing computational manipulations, starting from Fermat's principle, working through the theory of differential forms on manifolds and transferring these ideas to the applications of reduction by symmetry to reveal Lie–Poisson Hamiltonian formulations and momentum maps in physical applications. The many Exercises and Worked Answers aid the student to grasp the essential aspects of the subject. In addition, the modern language and application of differential forms is explained in the context of geometric mechanics, so that the importance of Lie derivatives and their flows is clear. All theorems are stated and proved explicitly. The book's many worked exercises make it ideal for both classroom use and self-study. In particular, a substantial appendix containing both introductory examples and enhanced coursework problems with worked answers is included to help the student develop proficiency in using the powerful methods of geometric mechanics.
Holm DD, 2008, Geometric mechanics: Part II: Rotating, translating and rolling, ISBN: 9781848161559
This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduate and beginning graduate students in mathematics, physics, and engineering. It treats the dynamics of rotating, spinning and rolling rigid bodies from a geometric viewpoint, by formulating their solutions as coadjoint motions generated by Lie groups. The only prerequisites are linear algebra, multivariable calculus and some familiarity with Euler-Lagrange variational principles and canonical Poisson brackets in classical mechanics at the beginning undergraduate level. Variational calculus on tangent spaces of Lie groups is explained in the context of familiar concrete examples. Through these examples, the student develops skills in performing computational manipulations, starting from vectors and matrices, working through the theory of quaternions to understand rotations, and then transferring these skills to the computation of more abstract adjoint and coadjoint motions, Lie-Poisson Hamiltonian formulations, momentum maps and finally dynamics with nonholonomic constraints. The 120 Exercises and 55 Worked Answers help the student to grasp the essential aspects of the subject, and to develop proficiency in using the powerful methods of geometric mechanics. In addition, all theorems are stated and proved explicitly. The book's many examples and worked exercises make it ideal for both classroom use and self-study.
Gibbon JD, Holm DD, 2008, Estimates for the LANS-α, Leray-α and Bardina Models in Terms of a Navier-Stokes Reynolds Number, INDIANA UNIVERSITY MATHEMATICS JOURNAL, Vol: 57, Pages: 2761-2773, ISSN: 0022-2518
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- Citations: 15
Graham JP, Holm DD, Mininni PD, et al., 2007, Highly turbulent solutions of the Lagrangian-averaged Navier-Stokes alpha model and their large-eddy-simulation potential, Physical Review E, Vol: 76, ISSN: 1539-3755
We compute solutions of the Lagrangian-averaged Navier-Stokes α- (LANS α) model for significantly higher Reynolds numbers (up to Re≈8300) than have previously been accomplished. This allows sufficient separation of scales to observe a Navier-Stokes inertial range followed by a second inertial range specific to the LANS α model. Both fully helical and nonhelical flows are examined, up to Reynolds numbers of ∼1300. Analysis of the third-order structure function scaling supports the predicted l3 scaling; it corresponds to a k−1 scaling of the energy spectrum for scales smaller than α. The energy spectrum itself shows a different scaling, which goes as k1. This latter spectrum is consistent with the absence of stretching in the subfilter scales due to the Taylor frozen-in hypothesis employed as a closure in the derivation of the LANS α model. These two scalings are conjectured to coexist in different spatial portions of the flow. The l3 [E(k)∼k−1] scaling is subdominant to k1 in the energy spectrum, but the l3 scaling is responsible for the direct energy cascade, as no cascade can result from motions with no internal degrees of freedom. We demonstrate verification of the prediction for the size of the LANS α attractor resulting from this scaling. From this, we give a methodology either for arriving at grid-independent solutions for the LANS α model, or for obtaining a formulation of the large eddy simulation optimal in the context of the α models. The fully converged grid-independent LANS α model may not be the best approximation to a direct numerical simulation of the Navier-Stokes equations, since the minimum error is a balance between truncation errors and the approximation error due to using the LANS α instead of the primitive equations. Furthermore, the small-scale behavior of the LANS α model contributes to a reduction of flux at constant energy, leading to a shallower ener
Holm DD, Putkaradze V, 2007, Formation and evolution of singularities in anisotropic geometric continua, PHYSICA D-NONLINEAR PHENOMENA, Vol: 235, Pages: 33-47, ISSN: 0167-2789
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- Citations: 9
Holm DD, Putkaradze V, Tronci C, 2007, Geometric dissipation in kinetic equations, COMPTES RENDUS MATHEMATIQUE, Vol: 345, Pages: 297-302, ISSN: 1631-073X
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- Citations: 7
Holm DD, Hook DW, Bender CM, 2007, Complexified dynamical systems., Journal of Mathematical Physics A - Mathematical and Theoretical, Vol: 40, Pages: F793-F804
Many dynamical systems, such as the Lotka- Volterra predator-prey model and the Euler equations for the free rotation of a rigid body, are PT symmetric. The standard and well-known real solutions to such dynamical systems constitute an infinitessimal subclass of the full set of complex solutions. This paper examines a subset of the complex solutions that contains the real solutions, namely those having PT symmetry. The condition of PT symmetry selects out complex solutions that are periodic
Gibbon JD, Holm DD, 2007, Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics, New Journal of Physics, Vol: 9, ISSN: 1367-2630
After a review of the isentropic compressible magnetohydrodynamics (ICMHD) equations, a quaternionic framework for studying the alignment dynamics of a general fluid flow is explained and applied to the ICMHD equations.
Gibbon JD, Holm DD, 2007, Lagrangian particle paths and ortho-normal quaternion frames, NONLINEARITY, Vol: 20, Pages: 1745-1759, ISSN: 0951-7715
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- Citations: 19
Holm DD, Kerr RM, 2007, Helicity in the formation of turbulence, Physics of Fluids, Vol: 19, ISSN: 1070-6631
Helicity in vortex structures and spectra is studied in the developmental stages of a numerical simulation of the Navier-Stokes equations using three-dimensional visualizations and spectra. First, time scales are set using the growth and decay of energy dissipation, the peak value of vorticity, and the helicity. Then, two stages between the early time, nearly inviscid Euler dynamics with vortex sheets and a final state of fully developed turbulence with vortex tubes, are described. In the first stage, helicity fluctuations develop in Fourier space during a period still dominated by vortex sheets and rapidly growing peak vorticity. At the end of this period the strongest structure consists of transverse vortex sheets with mixed signs of helicity. During the second stage, a dissipative interaction propagates along one of these vortices as the sheets roll each other into vortex tubes.
Bender CM, Holm DD, Hook DW, 2007, Complex trajectories of a simple pendulum., Journal of Physics A: Mathematical and Theoretical, Vol: 40, Pages: F81-F89
The motion of a classical pendulum in a gravitational field of strength g is explored. The complex trajectories as well as the real ones are determined. If g is taken to be imaginary, the Hamiltonian that describes the pendulum becomes PT-symmetric. The classical motion for this PT-symmetric Hamiltonian is examined in detail. The complex motion of this pendulum in the presence of an external periodic forcing term is also studied.
Cotter CJ, Holm DD, Hydon PE, 2007, Multisymplectic formulation of fluid dynamics using the inverse map, Proceedings of the Royal Society A, Vol: 463, Pages: 2671-2687
We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map, the ‘back-to-labels’ map, gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton's principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to Kelvin's circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.
Holm D, Gibbons J, Tronci C, 2007, Vlasov moments, integrable systems and singular solutions, Physics Letters A
BRODY D, ELLIS D, HOLM D, 2007, Random Hamiltonian in thermal equilibrium, Fourth International Workshop on Decoherence, Information, Complexity and Entropy
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