Geophysical research has focused on flows, such as ocean currents, as two dimensional. Two dimensional point or blob vortex models have the advantage of having a Hamiltonian, whereas 3D vortex filament or tube systems do not necessarily have one, although they do have action functionals. On the other hand, certain classes of 3D vortex models called nearly parallel vortex filament models do have a Hamiltonian and are more accurate descriptions of geophysical and atmospheric flows than purely 2D models, especially at smaller scales. In these quasi-2D'' models we replace 2D point vortices with vortex filaments that are very straight and nearly parallel but have Brownian variations along their lengths due to local self-induction. When very straight, quasi-2D filaments are expected to have virtually the same planar density distributions as 2D models. An open problem is when quasi-2D model statistics behave differently than those of the related 2D system and how this difference is manifested. In this paper we study the nearly parallel vortex filament model of Klein, Majda, Damodaran in statistical equilibrium. We are able to obtain a free-energy functional for the system in a non-extensive thermodynamic limit that is a function of the mean square vortex position @math and solve for @math . Such an explicit formula has never been obtained for a non-2D model. We compare the results of our formula to a 2-D formula of Lim:2005 and show qualitatively different behavior even when we disallow vortex braiding. We further confirm our results using Path Integral Monte Carlo (Ceperley (1995)) permutations and that the Klein, Majda, Damodaran model's asymptotic assumptions for parameters where these deviations occur.
As mentioned in the previous section, simulations of flux lines in type-II superconductors using the PIMC method have been done, generating the Abrikosov lattice ( @cite_13 , @cite_0 ). However, the superconductor model has periodic boundary conditions in the xy-plane, is a different problem altogether, and is not applicable to trapped fluids. No Monte Carlo studies of the model of @cite_18 have been done to date and dynamical simulations have been confined to a handful of vortices. @cite_5 added a white noise term to the KMD Hamiltonian, Equation , to study vortex reconnection in comparison to direct Navier-Stokes, but he confined his simulations to two vortices. Direct Navier-Stokes simulations of a large number of vortices are beyond our computational capacities.
@cite_0 @cite_5 @cite_18 @cite_13
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Using path integral Monte Carlo we simulate a 3D system of up to 1000 magnetic flux lines by mapping it onto interacting bosons in 2 1 1D. With increasing temperatures we find first order melting from an ordered solid to an entangled liquid signaled by a finite entropy jump and sharp discontinuities of the defect density and the structure factor SG. For a particular density of strong columnar pins the crystal is transformed into a Bose glass phase with patches of crystalline order disrupted by the trapped vortices at the pinning sites but with no overall positional or orientational order. This glassy phase melts into a defected entangled liquid through a continuous transition. We show that the stochastic differential equation (SDE) model for the merger of two identical two-dimensional vortices proposed by Agullo and Verga [“Exact two vortices solution of Navier–Stokes equation,” Phys. Rev. Lett. 78, 2361 (1997)] is a special case of a more general class of SDE models for N interacting vortex filaments. These toy models include vorticity diffusion via a white noise forcing of the inviscid equations, and thus extend inviscid models to include core dynamics and topology change (e.g., merger in two dimensions and vortex reconnection in three dimensions). We demonstrate that although the N=2 two-dimensional model is qualitatively and quantitatively incorrect, it can be dramatically improved by accounting for self-advection. We then extend the two-dimensional SDE model to three dimensions using the semi-inviscid asymptotic approximation of [“Simplified equations for the interactions of nearly parallel vortex filaments,” J. Fluid Mech. 288, 201 (1995)] for nearly parallel... We present an extensive numerical study of vortex matter using the mapping to two-dimensional bosons and path-integral Monte Carlo simulations. We find a ital first-order vortex lattice melting transition into an ital entangled vortex liquid. The jumps in entropy and density are consistent with experimental results on YBa sub 2 Cu sub 3 O sub 7 minus delta . The liquid is denser than the lattice and has a correlation length l sub z approx 1.7 var_epsilon a sub 0 in the direction parallel to the field. In the language of bosons we find a sharp quantum phase transition from a Wigner crystal to a superfluid, even in the case of logarithmic interaction. We also measure the excitation spectrum of the Bose system and find the roton minimum to be insensitive to the range of the interaction. copyright ital 1998 ital The American Physical Society