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Register a new userLagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics
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The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in the Lagrangian (variational) formulation. An alternative is the imposition of incompressibility in the Eulerian variable description by a generalization of Diracs constraint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility constraint using Diracs method in terms of both the canonical Poisson brackets in the Lagrangian variable description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of density. Both cases give dynamics of infinite-dimensional geodesic flow on the group of volume preserving diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various methods is made.
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Inspired by an approach proposed previously for the incompressible Navier-Stokes (NS) equations, we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics (MHD) on a torus of arbitrary dimension d; this setting involves a Sobolev space of infinite order, made of C^infinity vector fields (with vanishing divergence and mean) on the torus. Given any approximate solution of the MHD Cauchy problem, its a posteriori analysis with the method of the present work allows to infer a lower bound on the time of existence of the exact solution, and to bound from above the Sobolev distance of any order between the exact and the approximate solution. In certain cases the above mentioned lower bound on the time of existence is found to be infinite, so one infers the global existence of the exact MHD solution. We present some applications of this general scheme; the most sophisticated one lives in dimension d=3, with the ABC flow (perturbed magnetically) as an initial datum, and uses for the Cauchy problem a Galerkin approximate solution in 124 Fourier modes. We illustrate the conclusions arising in this case from the a posteriori analysis of the Galerkin approximant; these include the derivation of global existence of the exact MHD solution with the ABC datum, when the dimensionless viscosity and resistivity are equal and stay above an explicitly given threshold value.
We review the role of dual pairs in mechanics and use them to derive particle-like solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Taylor series, also known as jets. The augmented weak dual pairs induce a hierarchy of particle-like solutions and conservation laws with particles carrying a copy of a jet group. We call these augmented particles jetlets. The jet groups serve as finite-dimensional models of the diffeomorphism group itself, and so the jetlet particles serve as a finite-dimensional model of the self-similarity exhibited by ideal incompressible fluids. The conservation law associated to jetlet solutions is shown to be a shadow of Kelvins circulation theorem. Finally, we study the dynamics of infinite time particle mergers. We prove that two merging particles at the zeroth level in the hierarchy yield dynamics which asymptotically approach that of a single particle in the first level in the hierarchy. This merging behavior is then verified numerically as well as the exchange of angular momentum which must occur during a near collision of two particles. The resulting particle-like solutions suggest a new class of meshless methods which work in dimensions $n geq 2$ and which exhibit a shadow of Kelvins circulation theorem. More broadly, this provides one of the first finite-dimensional models of self-similarity in ideal fluids.
We demonstrate that, for the case of quasi-equipartition between the velocity and the magnetic field, the Lagrangian-averaged magnetohydrodynamics alpha-model (LAMHD) reproduces well both the large-scale and small-scale properties of turbulent flows; in particular, it displays no increased (super-filter) bottleneck effect with its ensuing enhanced energy spectrum at the onset of the sub-filter-scales. This is in contrast to the case of the neutral fluid in which the Lagrangian-averaged Navier-Stokes alpha-model is somewhat limited in its applications because of the formation of spatial regions with no internal degrees of freedom and subsequent contamination of super-filter-scale spectral properties. No such regions are found in LAMHD, making this method capable of large reductions in required numerical degrees of freedom; specifically, we find a reduction factor of 200 when compared to a direct numerical simulation on a large grid of 1536^3 points at the same Reynolds number.
Truncated Taylor expansions of smooth flow maps are used in Hamiltons principle to derive a multiscale Lagrangian particle representation of ideal fluid dynamics. Numerical simulations for scattering of solutions at one level of truncation are found to produce solutions at higher levels. These scattering events to higher levels in the Taylor expansion are interpreted as modeling a cascade to smaller scales.
We show that the matter Lagrangian of an ideal fluid equals (up to a sign -depending on its definition and on the chosen signature of the metric) the total energy density of the fluid, i.e. rest energy density plus internal energy density.
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