arXiv subjects

Darryl D. Holm

Publications and source records attributed to Darryl D. Holm.

At least 109 records · Page 6Linked to original sources

arXiv2009

Intermittency effects in rotating decaying turbulence

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).

nlin.CD↗

arXiv2009

The geodesic Vlasov equation and its integrable moment closures

Various integrable geodesic flows on Lie groups are shown to arise by taking moments of a geodesic Vlasov equation on the group of canonical transformations. This was already known for both the one- and two-component Camassa-Holm systems. The present paper extends our earlier work to recover another integrable system of ODE's that was recently introduced by Bloch and Iserles. Solutions of the Bloch-Iserles system are found to arise from the Klimontovich solution of the geodesic Vlasov equation. These solutions are shown to form one of the legs of a dual pair of momentum maps. The Lie-Poisson structures for the dynamics of truncated moment hierarchies are also presented in this context.

nlin.SI↗

arXiv2009

Dynamics of charged molecular strands

Euler-Poincare equations are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that the equations of motion for the dynamics of such molecular strands are nonlocal when the screened Coulomb interactions, or Lennard-Jones potentials between pairs of charges are included. These nonlocal dynamical equations are derived in the convective representation of continuum motion by using modified Euler-Poincare and Hamilton-Pontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton's principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods in the spatial representation. The motion equations in the convective representation are shown to be affine Euler-Poincare equations relative to a certain cocycle. This property relates the geometry of the molecular strands to that of complex fluids. An elegant change of variables allows a direct passage from the infinite dimensional point of view to the covariant formulation in terms of Lagrange-Poincare equations. In another revealing perspective, the convective representation of the nonlocal equations of molecular strand motion is transformed into quaternionic form.

nlin.AO↗

arXiv2009

Random Hamiltonian in thermal equilibrium

A framework for the investigation of disordered quantum systems in thermal equilibrium is proposed. The approach is based on a dynamical model--which consists of a combination of a double-bracket gradient flow and a uniform Brownian fluctuation--that `equilibrates' the Hamiltonian into a canonical distribution. The resulting equilibrium state is used to calculate quenched and annealed averages of quantum observables.

quant-ph↗

arXiv2008

Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions

The EPDiff equation (or dispersionless Camassa-Holm equation in 1D) is a well known example of geodesic motion on the Diff group of smooth invertible maps (diffeomorphisms). Its recent two-component extension governs geodesic motion on the semidirect product ${\rm Diff}\circledS{\cal F}$, where $\mathcal{F}$ denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on ${\rm Diff} \circledS\mathfrak{g}$, where $\mathfrak{g}$ denotes the space of scalar functions that take values on a certain Lie algebra (for example, $\mathfrak{g}=\mathcal{F}\otimes\mathfrak{so}(3)$). Measure-valued delta-like solutions are shown to be momentum maps possessing a dual pair structure, thereby extending previous results for the EPDiff equation. The collective Hamiltonians are shown to fit into the Kaluza-Klein theory of particles in a Yang-Mills field and these formulations are shown to apply also at the continuum PDE level. In the continuum description, the Kaluza-Klein approach produces the Kelvin circulation theorem.

nlin.SI↗

arXiv2008

Hamiltonian statistical mechanics

A framework for statistical-mechanical analysis of quantum Hamiltonians is introduced. The approach is based upon a gradient flow equation in the space of Hamiltonians such that the eigenvectors of the initial Hamiltonian evolve toward those of the reference Hamiltonian. The nonlinear double-bracket equation governing the flow is such that the eigenvalues of the initial Hamiltonian remain unperturbed. The space of Hamiltonians is foliated by compact invariant subspaces, which permits the construction of statistical distributions over the Hamiltonians. In two dimensions, an explicit dynamical model is introduced, wherein the density function on the space of Hamiltonians approaches an equilibrium state characterised by the canonical ensemble. This is used to compute quenched and annealed averages of quantum observables.

quant-ph↗

arXiv2008

Singular solutions of a modified two-component Camassa-Holm equation

The Camassa-Holm equation (CH) 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 dependence on average density as well as pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in 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 MCH2 are given and compared with the pure CH2 case. These numerics show that the modification in MCH2 to introduce average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for MCH2 shows a new asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, MCH2 also allows the phase shift of the peakon collision to diverge in certain parameter regimes.

physics.flu-dyn↗

arXiv2008

An Euler Poincaré framework for the multilayer Green Nagdhi equations

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 Poincare 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.

physics.flu-dyn↗

arXiv2008

The Euler-Poincare theory of Metamorphosis

In the pattern matching approach to imaging science, the process of ``metamorphosis'' is template matching with dynamical templates. Here, we recast the metamorphosis equations of into the Euler-Poincare variational framework of and show that the metamorphosis equations contain the equations for a perfect complex fluid \cite{Ho2002}. This result connects the ideas underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids. After developing the general theory, we reinterpret various examples, including point set, image and density metamorphosis. We finally discuss the issue of matching measures with metamorphosis, for which we provide existence theorems for the initial and boundary value problems.

cs.CV↗

arXiv2008

Geometric gradient-flow dynamics with singular solutions

The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy's Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.

nlin.AO↗

arXiv2008

Nonlocal orientation-dependent dynamics of molecular strands

Time-dependent Hamiltonian dynamics is derived for a curve (molecular strand) in $\mathbb{R}^3$ that experiences both nonlocal (for example, electrostatic) and elastic interactions. The dynamical equations in the symmetry-reduced variables are written on the dual of the semidirect-product Lie algebra $so(3) \circledS (\mathbb{R}^3\oplus\mathbb{R}^3\oplus\mathbb{R}^3\oplus\mathbb{R}^3)$ with three 2-cocycles. We also demonstrate that the nonlocal interaction produces an interesting new term deriving from the coadjoint action of the Lie group SO(3) on its Lie algebra $so(3)$. The new filament equations are written in conservative form by using the corresponding coadjoint actions.

nlin.AO↗

arXiv2007

Kinetic models of heterogeneous dissipation

We suggest kinetic models of dissipation for an ensemble of interacting oriented particles, for example, moving magnetized particles. This is achieved by introducing a double bracket dissipation in kinetic equations using an oriented Poisson bracket, and employing the moment method to derive continuum equations for magnetization and density evolution. We show how our continuum equations generalize the Debye-Hueckel equations for attracting round particles, and Landau-Lifshitz-Gilbert equations for spin waves in magnetized media. We also show formation of singular solutions that are clumps of aligned particles (orientons) starting from random initial conditions. Finally, we extend our theory to the dissipative motion of self-interacting curves.

nlin.AO↗

arXiv2007

Emergent singular solutions of non-local density-magnetization equations in one dimension

We investigate the emergence of singular solutions in a non-local 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 non-local effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of non-locality 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.

nlin.AO↗

arXiv2007

Geometric dissipation in kinetic equations

A new symplectic variational approach is developed for modeling dissipation in kinetic equations. This approach yields a double bracket structure in phase space which generates kinetic equations representing coadjoint motion under canonical transformations. The Vlasov example admits measure-valued single-particle solutions. Such solutions are reversible; and the total entropy is a Casimir, and thus is preserved.

physics.plasm-ph↗

arXiv2007

Applications of Poisson Geometry to Physical Problems

We consider Lagrangians in Hamilton's principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincaré equations. In this case, the invariant Lagrangian is defined on the Lie algebra of the group and its Euler-Poincaré equations are defined on the dual Lie algebra, where dual is defined by the operation of taking variational derivative. On the Hamiltonian side, the Euler-Poincaré equations are Lie-Poisson and they possess accompanying momentum maps, which encode both their conservation laws and the geometry of their solution space. The standard Euler-Poincaré examples are treated, including particle dynamics, the rigid body, the heavy top and geodesic motion on Lie groups. Additional topics deal with Fermat's principle, the $\mathbb{R}^3$ Poisson bracket, polarized optical traveling waves, deformable bodies (Riemann ellipsoids) and shallow water waves, including the integrable shallow water wave systems associated with geodesic motion on the diffeomorphisms. The lectures end with the semidirect-product Euler-Poincaré reduction theorem for ideal fluid dynamics. This theorem introduces the Euler--Poincaré variational principle for incompressible and compressible motions of ideal fluids, with applications to geophysical fluids. It also leads to their Lie-Poisson Hamiltonian formulation.

math.DS↗

arXiv2007

Complexified Dynamical Systems

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.

hep-th↗

arXiv2007

Implementation of the LANS-alpha turbulence model in a primitive equation ocean model

This paper presents the first numerical implementation and tests of the Lagrangian-averaged Navier-Stokes-alpha (LANS-alpha) turbulence model in a primitive equation ocean model. The ocean model in which we work is the Los Alamos Parallel Ocean Program (POP); we refer to POP and our implementation of LANS-alpha as POP-alpha. Two versions of POP-alpha are presented: the full POP-alpha algorithm is derived from the LANS-alpha primitive equations, but requires a nested iteration that makes it too slow for practical simulations; a reduced POP-alpha algorithm is proposed, which lacks the nested iteration and is two to three times faster than the full algorithm. The reduced algorithm does not follow from a formal derivation of the LANS-alpha model equations. Despite this, simulations of the reduced algorithm are nearly identical to the full algorithm, as judged by globally averaged temperature and kinetic energy, and snapshots of temperature and velocity fields. Both POP-alpha algorithms can run stably with longer timesteps than standard POP. Comparison of implementations of full and reduced POP-alpha algorithms are made within an idealized test problem that captures some aspects of the Antarctic Circumpolar Current, a problem in which baroclinic instability is prominent. Both POP-alpha algorithms produce statistics that resemble higher-resolution simulations of standard POP. A linear stability analysis shows that both the full and reduced POP-alpha algorithms benefit from the way the LANS-alpha equations take into account the effects of the small scales on the large. Both algorithms (1) are stable; (2) make the Rossby Radius effectively larger; and (3) slow down Rossby and gravity waves.

physics.ao-ph↗

arXiv2006

Formation and Evolution of Singularities in Anisotropic Geometric Continua

Evolutionary PDEs for geometric order parameters that admit propagating singular solutions are introduced and discussed. These singular solutions arise as a result of the competition between nonlinear and nonlocal processes in various familiar vector spaces. Several examples are given. The motivating example is the directed self assembly of a large number of particles for technological purposes such as nano-science processes, in which the particle interactions are anisotropic. This application leads to the derivation and analysis of gradient flow equations on Lie algebras. The Riemann structure of these gradient flow equations is also discussed.

nlin.AO↗