Publications
335 results found
Geurts BJ, Holm DD, Kuczaj AK, 2007, Coriolis induced compressibility effects in rotating shear layers, 11th EUROMECH European Turbulence Conference, Publisher: SPRINGER-VERLAG BERLIN, Pages: 383-385, ISSN: 0930-8989
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- Citations: 2
Holm DD, Putkaradze V, 2006, Formation of clumps and patches in self-aggregation of finite-size particles, PHYSICA D-NONLINEAR PHENOMENA, Vol: 220, Pages: 183-196, ISSN: 0167-2789
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- Citations: 61
Gibbon JD, Holm DD, 2006, Length-scale estimates for the LANS-α equations in terms of the Reynolds number, PHYSICA D-NONLINEAR PHENOMENA, Vol: 220, Pages: 69-78, ISSN: 0167-2789
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- Citations: 17
Graham JP, Holm DD, Mininni P, et al., 2006, Inertial range scaling, Karman-Howarth theorem, and intermittency for forced and decaying Lagrangian averaged magnetohydrodynamic equations in two dimensions, Physics of Fluids, Vol: 18, ISSN: 1070-6631
We present an extension of the Kármán-Howarth theorem to the Lagrangian averaged magnetohydrodynamic (LAMHD-α) equations. The scaling laws resulting as a corollary of this theorem are studied in numerical simulations, as well as the scaling of the longitudinal structure function exponents indicative of intermittency. Numerical simulations for a magnetic Prandtl number equal to unity are presented both for freely decaying and for forced two-dimensional magnetohydrodynamic (MHD) turbulence, solving the MHD equations directly, and employing the LAMHD-α equations at 1∕2 and 1∕4 resolution. Linear scaling of the third-order structure function with length is observed. The LAMHD-αequations also capture the anomalous scaling of the longitudinal structure function exponents up to order 8.
Gibbons J, Holm DD, Tronci C, 2006, Singular solutions for geodesic flows of Vlasov moments
The Vlasov equation for the collisionless evolution of the single-particleprobability distribution function (PDF) is a well-known example of coadjointmotion. Remarkably, the property of coadjoint motion survives the process oftaking moments. That is, the evolution of the moments of the Vlasov PDF is alsocoadjoint motion. We find that {\it geodesic} coadjoint motion of the Vlasovmoments with respect to powers of the single-particle momentum admits singular(weak) solutions concentrated on embedded subspaces of physical space. Themotion and interactions of these embedded subspaces are governed by canonicalHamiltonian equations for their geodesic evolution.
Geurts BJ, Holm DD, 2006, Commutator errors in large-eddy simulation, JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, Vol: 39, Pages: 2213-2229, ISSN: 0305-4470
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- Citations: 13
Cotter CJ, Holm DD, 2006, Discrete momentum maps for lattice EPDiff, Pages: 7-278
We focus on the spatial discretization produced by the VariationalParticle-Mesh (VPM) method for a prototype fluid equation the known as theEPDiff equation}, which is short for Euler-Poincar\'e equation associated withthe diffeomorphism group (of $\mathbb{R}^d$, or of a $d$-dimensional manifold$\Omega$). The EPDiff equation admits measure valued solutions, whose dynamicsare determined by the momentum maps for the left and right actions of thediffeomorphisms on embedded subspaces of $\mathbb{R}^d$. The discrete VPManalogs of those dynamics are studied here. Our main results are: (i) avariational formulation for the VPM method, expressed in terms of a constrainedvariational principle principle for the Lagrangian particles, whose velocitiesare restricted to a distribution $D_{\VPM}$ which is a finite-dimensionalsubspace of the Lie algebra of vector fields on $\Omega$; (ii) a correspondingconstrained variational principle on the fixed Eulerian grid which gives adiscrete version of the Euler-Poincar\'e equation; and (iii) discrete versionsof the momentum maps for the left and right actions of diffeomorphisms on thespace of solutions.
Holm DD, 2006, Peakons, Encyclopedia of Mathematical Physics: Five-Volume Set, ISBN: 9780125126601
Holm DD, Nitsche M, Putkaradze V, 2006, Euler-alpha and vortex blob regularization of vortex filament and vortex sheet motion, Journal of Fluid Mechanics, Vol: 555, Pages: 149-176, ISSN: 0022-1120
The Euler-alpha and the vortex blob model are two different regularizations of incom- pressible ideal fluid flow. Here, a regularization is a smoothing operation which controls the fluid velocity in a stronger norm than L2. The Euler-alpha model is the inviscid version of the Lagrangian averaged Navier - Stokes-alpha turbulence model. The vortex blob model was introduced to regularize vortex flows. This paper presents both models within one general framework, and compares the results when applied to planar and axisymmetric vortex filaments and sheets. By certain measures, the Euler-alpha model is closer to the unregularized flow than the vortex blob model. The differences that result in circular vortex filament motion, vortex sheet linear stability properties, and core dynamics of spiral vortex sheet roll-up are discussed. © 2006 Cambridge University Press.
Gibbon JD, Holm DD, Kerr RM, et al., 2006, Quaternions and particle dynamics in the Euler fluid equations, Nonlinearity, Vol: 19, Pages: 1969-1983, ISSN: 0951-7715
Cotter CJ, Holm DD, 2006, Singular solutions, momentum maps and computational anatomy, MFCA 2006 - International Workshop on Mathematical Foundations of Computational Anatomy, Pages: 18-28
Geurts BJ, Holm DD, 2006, Leray and LANS-α modelling of turbulent mixing, JOURNAL OF TURBULENCE, Vol: 7, Pages: 1-33, ISSN: 1468-5248
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- Citations: 46
Holm DD, Putkaradze V, 2005, Aggregation of finite-size particles with variable mobility, Physical Review Letters, Vol: 95, ISSN: 0031-9007
New model equations are derived for dynamics of aggregation of finite-size particles. The differences from standard Debye-Hückel and Keller-Segel models are that the mobility of particles depends on the configuration of their neighbors and linear diffusion acts on locally averaged particle density. The evolution of collapsed states in these models reduces exactly to finite-dimensional dynamics of interacting particle clumps. Simulations show these collapsed (clumped) states emerge from smooth initial conditions, even in one spatial dimension. Extensions to two and three dimenstions are also discussed.
Chen QN, Chen SY, Eyink GL, et al., 2005, Resonant interactions in rotating homogeneous three-dimensional turbulence, JOURNAL OF FLUID MECHANICS, Vol: 542, Pages: 139-164, ISSN: 0022-1120
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- Citations: 64
Holm DD, Wingate BIA, 2005, Baroclinic instabilities of the two-layer quasigeostrophic alpha model, Journal of Physical Oceanography, Vol: 35, Pages: 1287-1296, ISSN: 0022-3670
The class of alpha models for turbulence may be derived by applying Lagrangian averaging to the exact fluid equations and then making a closure approximation based on Taylor’s hypothesis of frozen-in fluctuations. This derivation provides a closed expression for the unknown pseudomomentum in the generalized Lagrangian mean theory of Andrews and McIntyre. In the current study, the mean effects of turbulence on baroclinic instability are explored, as determined by the two-layer quasigeostrophic-alpha model in quasigeostrophic (QG) balance. The QG-alpha model is found to lower the critical wavenumber, reduce the bandwidth of instability, and preserve the value of forcing at onset in the baroclinic case. It also preserves the fundamental dependence of baroclinic instability on the gradient of the potential vorticity. These results encourage using the alpha-model approach—based on combining Lagrangian averaging with Taylor’s hypothesis closure approximations—in simulations of global ocean circulation, because this class of turbulence closure models allows Lagrangian-averaged effects of baroclinic instability to be simulated on a coarse mesh.
Fabijonas BR, Holm DD, 2005, Elliptic instability in the Lagrangian-averaged Euler-Boussinesq-alpha equations, Physics of Fluids, Vol: 17, ISSN: 1070-6631
We examine the effects of turbulence on elliptic instability of rotating stratified incompressible flows, in the context of the Lagrangian-averaged Euler–Boussinesq-α (LAEB-α) model of turbulence. We find that the LAEB-α model alters the instability in a variety of ways for fixed Rossby number and Brunt–Väisälä frequency. First, it alters the location of the instability domains in the (γ,cosθ)-parameter plane, where θ is the angle of incidence the Kelvin wave makes with the axis of rotation and γ is the eccentricity of the elliptic flow, as well as the size of the associated Lyapunov exponent. Second, the model shrinks the width of one instability band while simultaneously increasing another. Third, the model introduces bands of unstable eccentric flows when the Kelvin wave is two dimensional. We introduce two similarity variables—one is a ratio of the Brunt–Väisälä frequency to the model parameter Υ0=1+α2β2 and the other is the ratio of the adjusted inverse Rossby number to the same model parameter. Here, α is the turbulence correlation length and β is the Kelvin wave number. We show that by adjusting the Rossby number and Brunt–Väisälä frequency so that the similarity variables remain constant for a given value of Υ0, turbulence has little effect on elliptic instability for small eccentricities (∣γ∣⪡1). For moderate and large eccentricities, however, we see drastic changes of the unstable Arnold tongues due to the LAEB-α model. Additionally, we find that introducing anisotropy in the vertical component of the transported velocity field merely alters the definition of the model parameter Υ0, which effectively reduces the original parameter value. When the similarity variables are viewed with the new definition, the results are similar to those for the isotropic case.
Cheskidov A, Holm DD, Olson E, et al., 2005, On a Leray-α model of turbulence, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 461, Pages: 629-649, ISSN: 1364-5021
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- Citations: 230
Holm DD, Marsden JE, 2005, Momentum maps and measure-valued solutions (Peakons, filaments, and sheets) for the EPDiff equation, Vol: 232, Pages: 203-235, ISSN: 0743-1643
This paper is concerned with the dynamics of measure-valued solutions of the EPDiff equations, standing for the Euler-Poincare equations associated with the diffeomorphism group (of ℝn or of an n-dimensional manifold M). It focuses on Lagrangians that are quadratic in the velocity fields and their first derivatives, that is, on geodesic motion on the diffeomorphism group with respect to a right invariant Sobolev H1 metric. The corresponding Euler-Poincaré (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations for shallow water waves in one dimension. The corresponding equations for the volume-preserving diffeomorphism group are the LAE (Lagrangian averaged Euler) equations for incompressible fluids.
Holm DD, Hone ANW, 2005, A class of equations with peakon and pulson solutions (with an appendix by Harry Braden and John Byatt-Smith), JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, Vol: 12, Pages: 380-394, ISSN: 1402-9251
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- Citations: 50
Cao C, Holm DD, Titi ES, 2005, On the Clark-α model of turbulence:: global regularity and long-time dynamics, JOURNAL OF TURBULENCE, Vol: 6, Pages: 1-11, ISSN: 1468-5248
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- Citations: 63
Fabijonas BR, Holm DD, 2004, Euler-Poincare formulation and elliptic instability for <i>n</i>th-gradient fluids, JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, Vol: 37, Pages: 7609-7623, ISSN: 0305-4470
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- Citations: 1
Cushman RH, Dullin HR, Giacobbe A, et al., 2004, CO2 molecule as a quantum realization of the 1 : 1 : 2 resonant swing-spring with monodromy, Physical Review Letters, Vol: 93, ISSN: 0031-9007
We consider the wide class of systems modeled by an integrable approximation to the 3 degrees of freedom elastic pendulum with 1∶1∶2 resonance, or the swing-spring. This approximation has monodromy which prohibits the existence of global action-angle variables and complicates the dynamics. We study the quantum swing-spring formed by bending and symmetric stretching vibrations of the CO2 molecule. We uncover quantum monodromy of CO2 as a nontrivial codimension 2 defect of the three dimensional energy-momentum lattice of its quantum states.
Fabijonas BR, Holm DD, 2004, Multi-frequency Craik-criminale solutions of the Navier-Stokes equations, JOURNAL OF FLUID MECHANICS, Vol: 506, Pages: 207-215, ISSN: 0022-1120
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- Citations: 2
Fabijonas BR, Holm DD, 2004, Craik-Criminale solutions and elliptic instability in nonlinear-reactive closure models for turbulence, Physics of Fluids, Vol: 16, Pages: 853-866, ISSN: 1070-6631
The Craik–Criminale class of exact solutions is examined for a nonlinear-reactive fluids theory that includes a family of turbulence closure models. These may be formally regarded as either large eddy simulation or Reynolds-averaged Navier–Stokes models of turbulence. All of the turbulence closure models in the class under investigation preserve the existence of elliptic instability, although they shift its angle of critical stability as a function of the rotation rate Ω of the coordinate system, the wave number β of the Kelvin wave, and the model parameter α, the turbulence correlation length. Elliptic instability allows a comparison among the properties of these models. It is emphasized that the physical mechanism for this instability is not wave–wave interaction, but rather wave, mean-flow interaction as governed by the choice of a model’s nonlinearity.
Dullin HR, Gottwald GA, Holm DD, 2004, On asymptotically equivalent shallow water wave equations, PHYSICA D-NONLINEAR PHENOMENA, Vol: 190, Pages: 1-14, ISSN: 0167-2789
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- Citations: 179
Holm DD, 2004, Peakons, Encyclopedia of Mathematical Physics: Five-Volume Set, Pages: 12-20, ISBN: 9780125126663
Peakons are singular solutions of the dispersionless Camassa-Holm (CH) shallow-water wave equation in one spatial dimension. These are reviewed in the context of asymptotic expansions and Euler-Poincaré (EP) variational principles. The dispersionless CH equation generalizes to the EPDiff equation (defined subsequently in this article), whose singular solutions are peakon wave fronts in higher dimensions. The reduction of these singular solutions of CH and EPDiff to canonical Hamiltonian dynamics on lower-dimensional sets may be understood, by realizing that their solution ansatz is a momentum map, and momentum maps are Poisson.
Holm DD, Putkaradze V, Stechman S, 2004, Rotating concentric circular peakons, Nonlinearity, Vol: 17, Pages: 2163-2186, ISSN: 0951-7715
Holm DD, Ratnanather JT, Trouvé A, et al., 2004, Soliton dynamics in computational anatomy, NEUROIMAGE, Vol: 23, Pages: S170-S178, ISSN: 1053-8119
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- Citations: 52
Holm DD, Marsden JE, Ratiu TS, 2004, The Euler-Poincare equations in geophysical fluid dynamics, Large-scale atmosphere-ocean dynamics : V. 2. Geometric methods and models, Editors: Norbury, Roulstone, Cambridge, Publisher: Cambridge University Press, Pages: 251-300, ISBN: 9780521807579
Holm DD, Putkaradze V, Weidman PD, et al., 2003, Boundary effects on exact solutions of the Lagrangian-averaged Navier-Stokes-α equations, Workshop on Progress in Statistical Hydrodynamics, Publisher: KLUWER ACADEMIC/PLENUM PUBL, Pages: 841-854, ISSN: 0022-4715
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- Citations: 10
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