statistical relevance of vorticity conservation in the hamiltonian particle mesh method we conduct long time simulations with a hamiltonian particle mesh method for ideal fluid flow to determine the statistical mean vorticity field of the discretization lagrangian and eulerian statistical models are proposed for the discrete dynamics and these are compared against numerical experiments the observed results are in excellent agreement with the theoretical models as well as with the continuum statistical mechanical theory for ideal fluid flow developed by ellis et al 2002 10 in particular the results verify that the apparently trivial conservation of potential vorticity along particle paths within the hpm method significantly influences the mean state as a side note the numerical experiments show that a nonzero fourth moment of potential vorticity can influence the statistical mean

statistical equilibrium measures and coherent states in two dimensional turbulence the equilibrium statistics of the euler equations in two dimensions are studied and a new continuum model of coherent or organized states is proposed this model is defined by a maximum entropy principle similar to that governing the miller robert model except that the family of global vorticity invariants is relaxed to a family of inequalities on all convex enstrophy integrals this relaxation is justified by constructing the continuum model from a sequence of lattice models defined by gibbs measures whose invariants are derived from the exact vorticity dynamics not a spectral truncation or spatial discretization of it the key idea is that the enstrophy integrals can be partially lost to vorticity fluctuations on a range of scales smaller than the lattice microscale while energy is retained in the larger scales a consequence of this relaxation is that many of the convex enstrophy constraints can be inactive in equilibrium leading to a simplification of the mean field equation for the coherent state specific examples of these simplified theories are established for vortex patch dynamics in particular a universal relation between mean vorticity and stream function is obtained in the dilute limit of the vortex patch theory which is different from the sinh relation predicted by the montgomery joyce theory of point vortices 1999 john wiley sons inc

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

minimal atmospheric finite mode models preserving symmetry and generalized hamiltonian structures abstract a typical problem with the conventional galerkin approach for the construction of finite mode models is to keep structural properties unaffected in the process of discretization we present two examples of finite mode approximations that in some respect preserve the geometric attributes inherited from their continuous models a three component model of the barotropic vorticity equation known as lorenz maximum simplification equations e n lorenz maximum simplification of the dynamic equations tellus 12 3 1960 243 254 and a six component model of the two dimensional rayleigh benard convection problem it is reviewed that the lorenz 1960 model respects both the maximal set of admitted point symmetries and an extension of the noncanonical hamiltonian form nambu form in a similar fashion it is proved that the famous lorenz 1963 model violates the structural properties of the saltzman equations and hence cannot be considered as the maximum simplification of the rayleigh benard convection problem using a six component truncation we show that it is again possible to retain both symmetries and the nambu representation in the course of discretization the conservative part of this six component reduction is related to the lagrange top equations dissipation is incorporated using a metric tensor

maximizing opto mechanical interaction using topology optimization this paper studies topology optimization of a coupled opto mechanical problem with the goal of finding the material layout which maximizes the optical modulation i e the difference between the optical response for the mechanically deformed and undeformed configuration the optimization is performed on a periodic cell and the periodic modeling of the optical and mechanical fields have been carried out using transverse electric bloch waves and homogenization theory in a plane stress setting respectively two coupling effects are included being the photoelastic effect and the geometric effect caused by the mechanical deformation for the studied objective and material choice it is concluded that the photoelastic effect and the geometric effect counteract each other which yields designs which are fundamentally different if the optimization takes only one effect into account when both effects are active a compromise is found however a strong regularization is needed in order to achieve reasonable 0 1 designs with a clear physical interpretation copyright 2011 john wiley sons ltd

finite dimensional lie poisson approximations to vlasov poisson equations many of the basic equations of conservative continuum mechanics euler vlasov poisson vlasov maxwell mhd etc are hamiltonian systems with respect to lie poisson brackets on dual spaces of infinite dimensional lie algebras the development of lie poisson integrators for finite dimensional lie poisson systems has shown that they are superior in the numerical simulations of these systems especially with regard to long term phenomena this paper shows how to truncate one of these systems the vlasov poisson equation of plasma physics to a finite dimensional lie poisson system this requires replacing the functions on single particle phase space with their poisson bracket lie algebra structure by a finite dimensional lie algebra replacing the densities by their moments of order up to k about a fixed reference point corresponds to replacing the functions by their taylor expansions up to order k unfortunately these truncated taylor expansions do not form a lie algebra since the functions which vanish through order k do not form an ideal under poisson bracket geometrically this corresponds to the fact that canonical transformations which fix the reference point do not form a normal subgroup introducing the location of a reference point in phase space as an extra variable and truncating with respect to this moving point turns out to decouple the location from shape coordinates of a lump of density in phase space as far as the poisson bracket is concerned one can then replace the shape coordinates by a finite number of moments the central result of the paper is a construction of the decoupling map described above in the general context of the decomposition of a lie group as a product of subgroups the main theorem is first proved by the general theory of poisson reduction then by explicit calculation and lastly by showing that the poisson isomorphism follows from the lift of a natural groupoid isomorphism the groupoid aspect of the theory also provides natural poisson maps useful in the application of ruth type integration techniques which do not seem easily derivable from the general theory of poisson reduction 1994 john wiley sons inc

numerical integration of lie poisson systems while preserving coadjoint orbits and energy in this paper we apply geometric integrators of the rkmk type to the problem of integrating lie poisson systems numerically by using the coadjoint action of the lie group g on the dual lie algebra mbox normalsize mathfrak g to advance the numerical flow we devise methods of arbitrary order that automatically stay on the coadjoint orbits first integrals known as casimirs are retained to machine accuracy by the numerical algorithm within the proposed class of methods we find integrators that also conserve the energy these schemes are implicit and of second order nonlinear iteration in the lie algebra and linear error growth of the global error are discussed numerical experiments with the rigid body and a finite dimensional truncation of the euler equations for a two dimensional 2d incompressible fluid are used to illustrate the properties of the algorithm

exact results on stationary turbulence in 2d consequences of vorticity conservation we establish a series of exact results for a model of stationary turbulence in two dimensions the incompressible navier stokes equation in a periodic domain with stochastic force white noise in time essentially all of our conclusions follow from the simple consideration of the simultaneous conservation of energy and enstrophy by the inertial dynamics our main results are as follows 1 we show the blow up of mean energy as l02e for 0 when there is no ir dissipation at the large length scale l0 2 with an additional ir dissipation we establish the validity of the traditional cascade directions and magnitudes of flux of energy and enstrophy for 0 assuming finite mean energy in the limit 3 we rigorously establish the balance equations for the energy and vorticity invariants in the 2d steady state and the forward cascade of the higher order vorticity invariants assuming finite mean values 4 we derive exact inequalities for scaling exponents in the 2d enstrophy range as follows if l p l p then 2 23 and p 0 for p 3 if the minimum holder exponent of the vorticity hmin 0 then we establish a 2d analogue of the refined similarity hypothesis which improves these bounds the most novel and interesting conclusion of this work is the connection established between intermittency in 2d and negative holder singularities of the vorticity we show that the latter are necessary for deviations from the 1967 kraichnan scaling to occur

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

sl 3 c generator matrix elements in a pauli subgroup basis we construct sl 3 c basis states reduced according to its finite subgroup 3 matrix elements of sl 3 c generators are calculated between 3 basis states

nested polyhedra model of turbulence a discretization of the wave number space is proposed using nested polyhedra in the form of alternating dodecahedra and icosahedra that are self similarly scaled this particular choice allows the possibility of forming triangles using only discretized wave vectors when the scaling between two consecutive dodecahedra is equal to the golden ratio and the icosahedron between the two dodecahedra is the dual of the inner dodecahedron alternatively the same discretization can be described as a logarithmically spaced with a scaling equal to the golden ratio nested dodecahedron icosahedron compounds a wave vector which points from the origin to a vertex of such a mesh can always find two other discretized wave vectors that are also on the vertices of the mesh which is not true for an arbitrary mesh thus the nested polyhedra grid can be thought of as a reduction or decimation of the fourier space using a particular set of self similar triads arranged approximately in a spherical form for each vertex i e discretized wave vector in this space there are either 9 or 15 pairs of vertices i e wave vectors with which the initial vertex can interact to form a triangle this allows the reduction of the convolution integral in the navier stokes equation to a sum over 9 or 15 interaction pairs transforming the equation in fourier space to a network of interacting nodes that can be constructed as a numerical model which evolves each component of the velocity vector on each node of the network this model gives the usual kolmogorov spectrum of k 5 3 since the scaling is logarithmic and the number of nodes for each scale is constant a very large inertial range i e a very high reynolds number with a much lower number of degrees of freedom can be considered incidentally by assuming isotropy and a certain relation between the phases the model can be used to systematically derive shell models

on the geometry of the hamilton jacobi equation and generating functions in this paper we develop a geometric version of the hamilton jacobi equation in the poisson setting specifically we geometrize what is usually called a complete solution of the hamilton jacobi equation we use some well known results about symplectic groupoids in particular cotangent groupoids as a keystone for the construction of our framework our methodology follows the ambitious program proposed by weinstein in mechanics day waterloo on 1992 volume 7 of fields institute communications american mathematical society providence 1996 in order to develop geometric formulations of the dynamical behavior of lagrangian and hamiltonian systems on lie algebroids and lie groupoids this procedure allows us to take symmetries into account and as a by product we recover results from channell and scovel phys d 50 1 80 88 1991 ge indiana univ math j 39 3 859 876 1990 ge and marsden phys lett a 133 3 134 139 1988 but even in these situations our approach is new a theory of generating functions for the poisson structures considered here is also developed following the same pattern solving a longstanding problem of the area how to obtain a generating function for the identity transformation and the nearby poisson automorphisms of poisson manifolds a direct application of our results gives the construction of a family of poisson integrators that is integrators that conserve the underlying poisson geometry these integrators are implemented in the paper in benchmark problems some conclusions current and future directions of research are shown at the end of the paper

mathematical analysis of a structure preserving approximation of the bidimensional vorticity equation we show the consistency and the convergence of a spectral approximation of the bidimensional vorticity equation proposed by v zeitlin in 13 and studied numerically by i szunyogh b kadar and d devenyi in 12 whose main feature is that it preserves the hamiltonian structure of the vorticity equation on demontre la consistance et la convergence d une approximation spectrale de l equation du tourbillon bidimensionnelle periodique proposee par v zeitlin dans 13 et etudiee numeriquement par i szunyogh b kadar et d devenyi dans 12 sa caracteristique principale est de preserver la structure hamiltonienne de l equation du tourbillon

complex trkalian fields and solutions to euler s equations for the ideal fluid we consider solutions to the complex trkalian equation vec nabla times vc vc where vc is a 3 component vector function with each component in the complex field and may be expressed in the form vc e ig vec nabla f with g real and f complex we find there are precisely two classes of solutions one where g is a cartesian variable and one where g is the spherical radial coordinate we consider these flows to be the simplest of all exact 3 d solutions to the euler s equation for the ideal incompressible fluid the novel approach we use in solving for these classes of solutions to these 3 dimensional vector pdes involves differential geometric techniques one may employ the method to generate solutions to other classes of vector pdes

negative temperature states and large scale long lived vortices in two dimensional turbulence we study onsager s theory of large coherent vortices in turbulent flows in the approximation of the point vortex model for two dimensional euler hydrodynamics in the limit of a large number of point vortices with the energy perpair of vortices held fixed we prove that the entropy defined from the microcanonical distribution as a function of the pair specific energy has its maximum at a finite value and thereafter decreases yielding the negative temperature states predicted by onsager we furthermore show that the equilibrium vorticity distribution maximizes an appropriate entropy functional subject to the constraint of fixed energy and under regularity assumptions obeys the joyce montgomery mean field equation we also prove that under appropriate conditions the vorticity distribution is the same as that for the canonical distribution a form of equivalence of ensembles we establish a large fluctuation theory for the microcanonical distributions which is based on a level 3 large deviations theory for exchangeable distributions we discuss some implications of that property for the ergodicity requirements to justify onsager s theory and also the theoretical foundations of a recent extension to continuous vorticity fields by r robert and j miller although the theory of two dimensional vortices is of primary interest our proofs actually apply to a very general class of mean field models with long range interactions in arbitrary dimensions

finite mode analogs of 2d ideal hydrodynamics coadjoint orbits and local canonical structure abstract the algebraic finite mode hydrodynamic type systems that have o n integrals of motion for o n n modes and are intrinsically connected with two dimensional ideal fluid flows are studied special emphasis is given on the structure of their phase space and the problem of finding proper dynamical variables for its description in connection with appropriate problems for the original hydrodynamic equations various possibilities for embedding in the space of canonical variables are demonstrated on the simplest example of the su 3 model giving the finite dimensional analogs of clebsch variables

on the statistical mechanics of 2d euler equation we address the issue of a rigorous justification of the statistical mechanics of 2d euler equation we construct a converging sequence of approximations of this equation for which a liouville theorem holds and such that the sequence of liouville measures has a large deviation property this provides an important step in the justification of the use of the entropy functional previously introduced in 8 11 13

invariants of the euler equations for ideal or barotropic hydrodynamics and superconductivity in d dimensions abstract the hamiltonian formalism for the euler equations of an ideal fluid superconductivity and a barotropic fluid on a d dimensional riemannian manifold is proposed we show that each of these equations has an infinite series of integrals if d is even generalized enstrophies and at least one integral if d is odd generalized helicity we prove that the magnetic hydrodynamics integral f v b is equal to the average linking number of vector fields rot v and b in terms of the ergodic theory all the invariants considered are casimir elements i e invariants of coadjoint action of the corresponding infinite dimensional lie algebras

momentum conserving symplectic integrators abstract in this paper we show that symplectic partitioned runge kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of hamiltonian differential equations by extended point transformation we also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of lie poisson systems on certain submanifolds of the general matrix group gl n