1988 Le Dizès and Verga 2002 Meunier et al. A numerical method is proposed in Rossi ( 1997, 2005) to capture these shape dynamics by adding basis functions with nontrivial moments in the study of vortex merger (see, for example, Melander et al. In particular, blobs of vorticity in real ideal fluids are known to change shape and deviate from initially circular distributions. While vortex blobs performed well, they did not capture all of the qualitative richness observed in fluid vorticity dynamics. ( 2006) for vortex filament and vortex sheet motion. A comparison of the Euler- α kernel to the m = 1 kernel of Beale and Majda ( 1985) is given in Holm et al. The Euler- α kernel is different from the kernels used in Chorin ( 1973) and Beale and Majda ( 1985). In particular, vortex blob solutions associated with a specific kernel serve as exact solutions to the Euler- α model (Oliver and Shkoller 2001). However, this thought was banished with the invention of the Euler- α model, a regularized model of ideal fluids with a parameter α representing the typical correlation length of fluctuations away from the mean of a Lagrangian fluid path (Foias et al. While this theoretical development clarified the geometry of point vortices, vortex blobs were sometimes thought to be a numerical “trick” which violated the geometric interpretation. The findings of Marsden and Weinstein ( 1983) were developed further in Gay-Balmaz and Vizman ( 2012) to handle fluid flow on manifolds with nontrivial homology. Simultaneously, the symplectic geometry of point vortices was clarified in Marsden and Weinstein ( 1983) by invoking Arnold’s interpretation of ideal fluids (Arnold 1966). Specifically, the convergence rate of the mth kernel was found to be of order h m q for any q ∈ (0, 1) where h = δ q is a grid-spacing parameter and δ > 0 is a length scale associated with the smoothing kernel (Beale and Majda 1982, 1985). By convolving the singular vortex kernel with sums of Gaussian smoothing kernels, a sequence of vortex blob kernels with faster convergence rates was found. Later, stronger convergence rates were achieved by judicious choice of vortex kernels. It was shown that the solutions of the vortex blob method converge to solutions of the Navier–Stokes equations in Hald ( 1979). In particular, the regularized system proved more amenable to error analysis. These adjustments to the classical point vortex method yielded the vortex blob method, which quickly became of practical use for realistic fluid flow modeling. Stochastic perturbations were further included to model viscosity (Chorin 1973).
In fact, the point vortex approach did not produce a competitive numerical method until the 1970s, when the problems related to singularities were overcome by regularizing the singular vortex kernel to form a vortex blob. At the beginning of their development, the infinite velocities (and energies) associated with point vortices caused great difficulties, both numerically and theoretically.
For example, the use of point vortices as idealized solutions can already be found in a 1931 paper concerning a “line of discontinuity” in planar fluid flow (Rosenhead 1931). Vortex methods for fluid modeling predate the computer age, and references to them can be found in the work of Helmholtz (Smith 2011, see the introductory section). Such findings illuminate the rich dynamics which occur below the regularization length scale and enlighten our perspective on the potential for regularized fluid models to capture multiscale phenomena. Applications to the design of numerical methods similar to vortex blob methods are also discussed. We also describe the symplectic geometry associated with these augmented vortex structures, and we characterize the dynamics as Hamiltonian. We determine the conditions under which such regularized Euler fluid equations may admit vorticity singularities which are stronger than delta functions, e.g., derivatives of delta functions. More specifically, a new singular vortex theory is presented for regularized Euler fluid equations of ideal incompressible flow in the plane. In this article, we observe that the dynamics need not be trivial if one is willing to consider distributional derivatives of Dirac delta functionals as valid vorticity distributions. Vortex blob methods are typically characterized by a regularization length scale, below which the dynamics are trivial for isolated blobs.
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