Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics
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Abstract
Vortex blob methods are typically characterized by a regularization length scale, below which the dynamics are trivial for isolated blobs. 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. More specifically, a new singular vortex theory is presented for regularized Euler fluid equations of ideal incompressible flow in the plane. 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. We also describe the symplectic geometry associated with these augmented vortex structures, and we characterize the dynamics as Hamiltonian. Applications to the design of numerical methods similar to vortex blob methods are also discussed. 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.
Keywords
Vortex blob methods Singular momentum maps Regularized Euler fluid equations Hamiltonian dynamicsMathematics Subject Classification
76M23 76M60 70H151 Introduction
A point vortex approximation to a continuous distribution of vorticity for Euler’s fluid equations is problematic, though a point vortex induces a flow velocity which becomes unbounded. However, when the point vortex is made smooth and bounded (regularized), the approximation becomes reasonable (Chorin 1973).
The economy of the vortex blob method derives from the property that Dirac delta distributions are hyperlocal (i.e., parametrized by position), and the property that the vorticity equation (1) admits Dirac delta distributions as solutions. However, there are many distributions which are localized to a similar degree (e.g., derivatives of delta functions, \(\partial _x \delta _{z_i}\)).
1.1 Main Contributions
 (1)
Section 2 briefly reviews the background for vortex methods in fluid modeling.
 (2)
Section 3 reviews the relationship between regularized fluids and vortex blob methods.
 (3)
Section 4 derives the equations of motion for point vortices and MVBs as exact solutions of a regularized vorticity equation.
 (4)
Section 5 derives the conservation laws for these equations, such as energy, linear momentum, and angular momentum, and circulation. The derivation of these conserved quantities as symplectic momentum maps can be found in Appendix B.
 (5)
Section 6 explains the relationship between the dynamical systems for MVBs and an implicitly defined closed dynamical system which governs the spatial moments of the vorticity distribution.
 (6)
Section 7 discusses numerical aspects of using MVBs to model fluid dynamics, such as approximations of initial conditions (Sect. 7.2), and grouping of computational nodes (Sect. 7.1).
 (7)
Section 8 presents the results of several numerical experiments involving small numbers of vortices, for \(N=1,2\), and 3.
 (8)
MVB dynamics are Hamiltonian. We present the symplectic and Hamiltonian structure of MVB dynamics in Sect. 9.
2 Background
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). 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). At the beginning of their development, the infinite velocities (and energies) associated with point vortices caused great difficulties, both numerically and theoretically. 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. Stochastic perturbations were further included to model viscosity (Chorin 1973). These adjustments to the classical point vortex method yielded the vortex blob method, which quickly became of practical use for realistic fluid flow modeling. In particular, the regularized system proved more amenable to error analysis. It was shown that the solutions of the vortex blob method converge to solutions of the Navier–Stokes equations in Hald (1979). Later, stronger convergence rates were achieved by judicious choice of vortex kernels. By convolving the singular vortex kernel with sums of Gaussian smoothing kernels, a sequence of vortex blob kernels with faster convergence rates was found. Specifically, the convergence rate of the mth kernel was found to be of order \(h^{mq}\) for any \(q \in (0,1)\) where \(h = \delta ^q\) is a gridspacing parameter and \(\delta > 0\) is a length scale associated with the smoothing kernel (Beale and Majda 1982, 1985).
Simultaneously, the symplectic geometry of point vortices was clarified in Marsden and Weinstein (1983) by invoking Arnold’s interpretation of ideal fluids (Arnold 1966). The findings of Marsden and Weinstein (1983) were developed further in GayBalmaz and Vizman (2012) to handle fluid flow on manifolds with nontrivial homology. 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. However, this thought was banished with the invention of the Euler\(\alpha \) model, a regularized model of ideal fluids with a parameter \(\alpha \) representing the typical correlation length of fluctuations away from the mean of a Lagrangian fluid path (Foias et al. 2001). In particular, vortex blob solutions associated with a specific kernel serve as exact solutions to the Euler\(\alpha \) model (Oliver and Shkoller 2001). The Euler\(\alpha \) kernel is different from the kernels used in Chorin (1973) and Beale and Majda (1985). A comparison of the Euler\(\alpha \) kernel to the \(m=1\) kernel of Beale and Majda (1985) is given in Holm et al. (2006) for vortex filament and vortex sheet motion.
While vortex blobs performed well, they did not capture all of the qualitative richness observed in fluid vorticity dynamics. In particular, blobs of vorticity in real ideal fluids are known to change shape and deviate from initially circular distributions. 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. 1988; Le Dizès and Verga 2002; Meunier et al. 2005). Another distinct model obtained by projection onto a Hermite basis is described in Nagem et al. (2009). This projection yielded a finite dimensional systems which modeled the (truncated) moments of the vorticity of an ideal incompressible fluid. The derivation of simplified combinatorial formulas invoked by the dynamics of this model was discovered in Uminsky et al. (2012), and these formulas have made the method numerically tractable for a large number of moments.
A dual approach to the momentbased methods of the previous paragraph (Rossi 1997, 2005; Nagem et al. 2009) is to consider multipolebased methods. This is the approach proposed in Nicolaides (1986), where an initial vortex ansatz consisting of sums of distributional derivatives of dirac delta distributions is considered. Such an idea has occurred intermittently in various forms in the literature, over many years. For example, a regularized vortex blob model, in the spirit of Beale and Majda (1982, 1985), which considered vorticity distributions of the form \(\omega = \sum \Gamma _i \delta _{z_i} + \Gamma _i^x \partial _x \delta _{z_i} + \Gamma _i^y \partial _y \delta _{z_i}\) was investigated in Chiu and Nicolaides (1988). Here it was proven that this augmentation of the traditional vortex method will yield faster spectral convergence than that of a traditional vortex blob method. The current article considers higher order derivatives and can be seen as a natural followup to (Chiu and Nicolaides 1988). More recently, dynamics have been derived for interactions of pure vortices and pure dipoles. These come from vorticity distributions of the form \(\omega = \sum _{i} \Gamma _i \delta _{z_{v,i}} + \sum _{j} \left( \Gamma _i^x \partial _x \delta _{z_{d,i}} + \Gamma _i^y \partial _y \delta _{z_{d,i}}\right) \) with the assumption that the locations of the dipoles, and the vortices never overlap and that their selfinteraction terms may be neglected (Yanovsky et al. 2009; Tur et al. 2011). In a different approach, approximations of dipoles are created by holonomically constraining vortices of opposite strength to be a fixed distance from one another (Tchieu et al. 2012). The question remains, however, to what extent the dynamics of Tchieu et al. (2012) approximates those of Yanovsky et al. (2009) and Tur et al. (2011) after selfinteraction terms have been neglected. In summary, the removal of selfinteraction terms is one of the primary obstacles to obtaining a multipolebased generalization of the point vortex method (Smith 2011). Moreover, the spectral error decay rates found in Hald (1979), Beale and Majda (1982, 1985) and Chiu and Nicolaides (1988) arise from the use of vortex blobs in place of (singular) point vortices. In this article, we will follow this regularizationbased approach.
3 Vortex Blobs and Regularized Fluid Models
It is notable that (9) and (7) together form a finite dimensional ODE. The solutions of this ODE are exact solutions to the regularized fluid model. Again, this is valuable because the solutions of many regularized fluid models have been shown to converge to solutions of the ideal fluid equations as \(\alpha \) vanishes. This paper seeks to generalize these pointlike solutions to regularized fluid models to obtain a richer class of solutions with richer conservation properties.
4 Equations of Motion
In this section, we derive the equations of motion for the timedependent parameters which specify multipole vortex blobs (MVBs). The zerothorder MVBs are just standard vortex blobs, and the resulting equations of motion are those of the standard (nonstochastic) vortex blob algorithm (Chorin 1973). The firstorder MVBs are regularized dipoles, and the equations of motion are those of Chiu and Nicolaides (1988). Here we will derive the equations of motion for Nthorder MVBs following the approach of Chiu and Nicolaides (1988).
We seek equations of motion for the \(\Gamma ^{mn}_i(t)\)’s and \(z_i(t)\)’s such that the velocity field (10) satisfies the vorticity equation (1). In the following calculations, we will not show the explicit time dependence of the dynamical variables.
Remark 4.1
5 Conserved Quantities
As (1) is a Hamiltonian system, we can consider searching for symmetries to find other conserved quantities using Noether’s theorem. We’ve relegated the discussion of the relevant symplectic structure to Appendix 2, where derivations and proofs of the following can be found. Here we can summarize the appendix.
6 Moments
In this section, we present how the moments of the vorticity distribution evolve in time. We will find that when the vorticity distribution is that of a MVB, and then the moments form a closed dynamical system at finite order.
Remark 6.1
This relation between the \(\Gamma \)’s and the \(\mu \)’s may also be important in the context of plasma physics, especially when one recalls that (1) can be interpreted as a onedimensional plasma model. Specifically, phasespace moments of the Vlasov probability distribution form an important dynamical link between Lagrangianparticle and Euleriancontinuum descriptions. The phasespace moments of the Vlasov probability distribution provide collective coordinates for the Hamiltonian dynamics of ensembles of particles. For more explanation of this property of Hamiltonian collectivization of the phasespace moments, see Guillemin and Sternberg (1990), Holm et al. (1990) and Gibbons et al. (2008a, b). In plasma dynamics, the phasespace moments arise from a Taylor expansion of the Vlasov particle distribution, taken around its centroid in phasespace. For planar incompressible flow of an ideal fluid, the phasespace comprises the (x, y) Lagrangian coordinates of a fluid particle, and the corresponding moments arise from Taylor expansions around the centroid of the (smooth) vorticity distribution. The duality between the resulting spatial moments of a smooth vorticity distribution and the MVBs corresponding to higher order singular vorticity distributions also obtained from a Taylor expansion raises the intriguing question of finding a relation between these two types of dynamical description. This question is particularly intriguing because the dynamics of moments beyond quadratic order in general does not close to form a finite dimensional Hamiltonian system, while the dynamics of MVBs closes at every order.
Remark 6.2
There exist other systems for approximating the dynamics of moments which differ from the one presented here. In particular, the equations of motion for the moments here form a closed system at order N, whereas other methods for deriving dynamical systems for moments (Uminsky et al. 2012; Nagem et al. 2009; Gibbons et al. 2008a, b) require truncations in order to form a closed system. For example, Uminsky et al. (2012) approximates the stream function as a sum of Hermite functions with evolving centroids and weights. In order to obtain the evolution for the weights and the centroids, they project the viscous vorticity equation onto this space via \(L^2\) projection. The resulting formulas are explicit and efficient to compute, albeit more complex than the formulas found in this paper. The primary source of error for Uminsky et al. (2012) over long times is the discrepancy between the projected evolution equations and the true evolution equations. In contrast, we approximate an Euler fluid with a regularized fluid equation which we solve exactly. This is not to say that error is not accumulated in time. The primary source of error for our method over long times is the discrepancy between the regularized fluid equations and the true fluid equations.
Admittedly, the equations of motion for the moments in Uminsky et al. (2012) bear some resemblance to the equations of motion for the \(\Gamma \)’s in our method. Both are quadratic in their respective variables, with coefficients involving combinatorial functions. A more precise relationship, if one exists, is difficult to discern. Philosophically, the methods share much in common. However, due to the fundamental approximation technique of projecting the equations of motion versus regularizing them, the methods are indeed distinct. This difference cascades throughout the study of both methods. For example, the convergence for Uminsky et al. (2012) is obtained via the convergence of spectral approximations, while the convergence of our method is a corollary of the convergence of a regularized fluid model (see Mumford and Michor 2013; Foias et al. 2001 for such convergence proofs).
7 Numerical Aspects
In this section, we discuss various numerical aspects of using MVBs to model fluid dynamics. We will observe how MVBs can be used to reduce the number of necessary pairwise computations without a drastic compromise in accuracy. We will also present an algorithm for constructing an initial condition of MVBs from a given stream function.
Remark 7.1
We refer to Chiu and Nicolaides (1988) for a convergence proof and error analysis of the firstorder case. A convergence proof is beyond the scope of this article. Suffice it to say, such a proof would likely resemble Chiu and Nicolaides (1988).
7.1 Grouping and Reduction of Pairwise Computations
Remark 7.2
Such reductions are even more dramatic when considering higher order jets. In particular, \(2^N\) zerothorder MVBs can be approximated with a single Nthorder MVB by applying the above approximations iteratively.
The computation of pairwise interactions in the vortex method was once a major bottleneck in implementing the standard vortex method for realworld applications. It was not until the invention of the fast multipole method that it became tractable to compute millions of pairwise interactions by reducing the complexity from an \({\mathcal {O}}(n^2)\) calculation to an \({\mathcal {O}}(n \log (n))\) calculation, where n is the number of vortices (Greengard and Rokhlin 1987). However, in the case of viscous fluids with boundaries, vorticity is shed from the boundaries. As a result, the vortex blob method of Chorin (1973) created new vortices at the boundary by using the Kutta condition as a creation criteria. For these applications, n will grow in time without bound, and some means of discarding vortices must be invoked. It is here that the grouping of MVBs could be useful. If one merges two Nthorder MVBs to obtained a \((N+1)\)thorder MVB, the amount of scalars and data typically increases. So one must still make a tough decision as to what data to discard (e.g., through some tolerance or by simply truncating at level M). Nonetheless, the analysis presented here could shed light on how best to implement this approach.
Remark 7.3
The merging of blobs of vorticity has been studied analytically (Melander et al. 1988) and numerically (Weiss and McWilliams 1993; Melander et al. 1988; Le Dizès and Verga 2002), as well as in the laboratory (Fine et al. 1991). All of this study has been in the slightly viscous (or nearly inviscid) regime. The grouping approach discussed here can be used to numerically resolve such collision events. In theory, there is no issue with collisions because we are considering regularized vortices where the induced velocity field from a single MVB is always finite. However, as \(\delta \) becomes smaller, the velocity near the vortex core diverges. This should be of concern as the convergence analysis of the vortex method presupposes that \(\delta \ll 1\). Typically such a near collision is handled by using a smaller time step (as the ODE is quite stiff). Grouping of MVBs suggests an alternative by avoiding this pairwise interaction altogether. Perhaps such an approach could be viewed as a variation of the punctuated dissipation events described in Weiss and McWilliams (1993) where an initial vorticity distribution is found to asymptotically approach a smoother axisymmetric vortex blob, and discrete vortex mergers are implemented to model this behavior.
Remark 7.4
There are qualitative questions which arise from mergers. For example, when two zerothorder vortex blobs are near each other, they will typically scatter after some finite time. Merging these blobs into a single firstorder blob will prohibit this scattering event from ever occurring. That both the zerothorder MVB solution and the merged firstorder MVB represent exact solutions of the fluid (after the merger event) are attributable to the longterm sensitivity to initial conditions near collision events. The scattering angle can be virtually anything since zerothorder MVBs can waltz around each other many times before scattering. The amount of time two zerothorder MVBs can spend waltzing around each other, and perhaps the merged solution represent some sort of limiting solution. That is to say, the merged solutions can be interpreted as the “waltzing for eternity” solution.
The irreversibility of merging is disturbing when one takes it to its extreme, one massive high order MVB. In order to address this, a means of splitting high order MVBs into lower order ones should be considered. The primary difficulty here is in determining when to split. In the case of mergers, we can decide to merge MVBs when they are close. Such a criterion is not immediately apparent in the case of splitting MVBs.
7.1.1 A Numerical Experiment with Grouping
7.2 Approximation of Initial Conditions
Let K be a compact set, and let \(0 < h \ll 1\) be small so that we may define the finite grid \(\Lambda _{h} = \{ (ah,bh) \in K \mid (a,b) \in \mathbb {Z}^2 \}\).^{1} Given an \(\omega \in {\mathcal {D}}'({\mathbb {R}}^2)\), we can attempt to approximate \(\omega \) via Dirac deltas supported on \(h \mathbb {Z}^2\). There is a natural way to do this with respect to the inner product \(\langle \cdot , \cdot \rangle _{G_\delta }\). We could define \(\omega _h^{(0)} = \sum _{i \in \mathbb {Z}^2} \Gamma _i \delta _{z_i}\) by requiring the error, \(\omega _h^{(0)}  \omega \), to be \(\langle \cdot , \cdot \rangle _{G_\delta }\)orthogonal to \(\delta _z\) for each \(z \in \Lambda _{h}\). This means that \(G_\delta *\omega (z) = \sum _i \Gamma _i G_\delta (zz_i)\) for each \(z \in \Lambda _{h}\). Thus, \(\psi _h^{(0)} = \sum _i \Gamma _i G_\delta (zz_i)\) can be seen as a zerothorder approximation to \(\psi = G_\delta *\omega \) because \(\psi _h^{(0)}(z) = \psi (z)\) for all \(z \in \Lambda _{h}\). Therefore, for smooth \(\omega \)’s, we obtain an error of order \({\mathcal {O}}( \Delta x)\) for a grid spacing of \(\Delta x\) using zerothorder MVBs.
In terms of complexity, in order to achieve a desired error bound, \(e_{tol} >0 \), you would need to use a grid with \({\mathcal {O}}( e_{tol}^{2/(k+1)} )\) MVBs. While the number of MVBs drops as k increases, one could object that a high order MVB is much more complex than a low order one. However, the number of degrees of freedom for a kthorder MVB is \(2 + \sum _{j=0}^{k} (2^k / k!)\) which monotonically converges to a constant (roughly 9.39) as \(k \rightarrow \infty \). Therefore, the number of degrees of freedom is dominated by \({\mathcal {O}}( e_{tol}^{2/(k+1)} )\) as well. In other words, when \(\psi \) is highly differentiable we observe benefits in terms of complexity and storage to using a larger k regardless of whether one measures complexity by the number of parameters to keep track of, or the number of MVBs.
8 Numerical Experiments
In these section, we present the results of numerical experiments involving small numbers of vortices, for \(N=1,2\), and 3.
8.1 Behavior of Isolated MVBs
Next, we will briefly explore the behavior of a single isolated kthorder MVB with \(\Gamma ^{mn} = 0\) with \(m+n < k\) for \(k=0,1,2\). This case allows us to investigate the dynamics induced by the higher order circulation variables in the absence of the lower order ones.
8.2 Order 0
The behavior of a single zerothorder MVB is explicitly solvable because the dynamics are stationary.
8.3 Order 1
8.4 Order 2
8.5 A Scattering Experiment
8.6 The Method of Images
9 Hamiltonians and Symplectic Structures
Having identified a symplectic manifold, \({{\mathrm{Orb}}}(\omega _0)\), we can then ask the question “are the dynamics Hamiltonian on \({{\mathrm{Orb}}}(\omega _0)\)?” Of course, the answer is “yes”. This is the primary content of Marsden and Weinstein (1983). We provide our own explanation here for convenience.
9.1 The FirstOrder Case
Remark 9.1
The use of this symplectic structure shows that the map \((z_i, \Gamma _i,\Gamma _i^x,\Gamma _i^y) \mapsto \omega \in {{\mathrm{Orb}}}(\omega )\) is a symplectic momentum map.
Remark 9.2
\(\{ \cdot , \cdot \}\)  x  y  \(\Gamma \)  \(\Gamma ^x\)  \(\Gamma ^y\) 

x  0  1  0  0  1 
y  −1  0  0  −1  0 
\(\Gamma \)  0  0  0  0  0 
\(\Gamma ^x\)  0  −1  0  0  1 
\(\Gamma ^y\)  1  0  0  −1  0 
10 Conclusion
In this paper, we have considered a generalization of the standard vortex blob method, obtained by augmenting the vortices with higher order circulation variables and dubbing them multipole vortex blobs (MVBs). By viewing the vorticity equation as an advection equation, we have obtained equations of motion for these MVBs.
The extra degrees of freedom of MVBs resulted in richer dynamics near the vortex core. Moreover, these new vorticity carrying elements exhibited a variety of novel types of solution behavior. We also observed faster convergence rates in space using higher order MVBs. Moreover, we proposed a scheme to decrease the number of pairwise interactions, by grouping MVBs of lower order into a smaller number of MVBs of higher order. Lastly, the implications of Kelvin’s circulation theorem were substantially richer in the case of MVBs than they were for the standard vortex blob method.
We have demonstrated the behavior of the MVBs with a sequence of simple numerical experiments consisting of small numbers of MVBs of various degrees. We found that firstorder MVBs correspond to sums of vortex blobs and regularized dipoles which simply propagate themselves forward, while the secondorder circulation variables activate richer (nonpropagating) dynamics near the vortex core.
Finally, we derived the symplectic structure of MVBs using methods from Marsden and Weinstein (1983). The resulting structure turned out to be a direct generalization of the standard symplectic structure for vortex blobs.
The multiscale nature of ideal fluids is the principal obstacle to obtaining accurate models (Chorin 1994, Chapter 3). The use of MVBs augments the standard vortex blob method by allowing for singular vorticity distributions which model dynamics below the regularization length scale (i.e., at order \(\delta ^k\) with \(\delta \ll 1\) for a kthorder jet vortex). As the dynamics of MVBs are relatively easy to derive, and their analysis is tractable, we believe that MVBs will be of considerable value in understanding the place of regularized fluid models within the computational fluids community at large and they should provide renewed interest in the vortex blob method.

MVBs on manifolds, such as the sphere

The convergence properties of the MVB method

How does one choose the regularization length scale in relation to the grid resolution. This relationship is addressed quite well for zerothorder MVBs in Beale and Majda (1982). It is not clear if higher order MVBs change those results.

An investigation of the kinetic theory of MVBs.
Footnotes
 1.
The choice of K should depend on the initial circulation \(\omega \), e.g., if the \(\omega \) has compact support than any K which contains the support of \(\omega \) would be a good candidate. Nonetheless, having to choose K is a weakness of the given approximation procedure.
Notes
Acknowledgements
Both authors gratefully acknowledge partial support by the European Research Council Advanced Grant 267382 FCCA to DDH. We thank Anatoly Tur and Vladimir Yanovsky for helping us navigate the literature and relate our paper to earlier work. We also thank Stefan Llewellyn Smith for his helpful comments before our initial submission.
References
 Arnold, V .I., Khesin, B .A.: Topological Methods in Hydrodynamics, Applied Mathematical Sciences. Springer, New York (1992)MATHGoogle Scholar
 Abraham, R., Marsden, J.E.: Foundations of Mechanics, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., Second edition, revised and enlarged, with the assistance of Tudor Ratiu and Richard Cushman. Reprinted by AMS Chelsea, 2008 (1978)Google Scholar
 Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 316–361 (1966)CrossRefGoogle Scholar
 Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, London (2000)Google Scholar
 Beale, J.T., Majda, A.: Vortex methods. II. Higher order accuracy in two and three dimensions. Math. Comp. 39(159), 29–52 (1982)MathSciNetMATHGoogle Scholar
 Beale, J.T., Majda, A.: Highorder accurate vortex methods with explicit velocity kernels. J. Comput. Phys. 58, 188–208 (1985)CrossRefMATHGoogle Scholar
 Chorin, A.: A numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–796 (1973)MathSciNetCrossRefGoogle Scholar
 Chorin, A.: Vorticity and Turbulence, Applied Mathematical Sciences, vol. 103. Springer, London (1994)CrossRefGoogle Scholar
 Chiu, C., Nicolaides, R.A.: Convergence of a higherorder vortex method for twodimensional Euler equations. Math. Comp. 51(184), 507–534 (1988)MathSciNetCrossRefMATHGoogle Scholar
 Constantine, G.M., Savits, T.H.: A multivariate Faà di Bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503–520 (1996)CrossRefMATHGoogle Scholar
 Fine, K.S., Driscoll, C.F., Malmberg, J.H., Mitchell, T.B.: Measurements of symmetric vortex merger. Phys. Rev. Lett. 67, 588–591 (1991)CrossRefGoogle Scholar
 Foias, C., Holm, D.D., Titi, E.S.: The Navier–Stokesalpha model of fluid turbulence. Phys. D 152–153, 505–519 (2001)MathSciNetCrossRefMATHGoogle Scholar
 GayBalmaz, F., Vizman, C.: Dual pairs in fluid dynamics. Ann. Glob. Anal. Geom. 41(1), 1–24 (2012)MathSciNetCrossRefMATHGoogle Scholar
 Gibbons, J., Holm, D.D., Tronci, C.: Singular solutions for geodesic flows of Vlasov moments. Probability, geometry and integrable systems, Math. Sci. Res. Inst. Publ., vol. 55, Cambridge Univ. Press, Cambridge, 2008, pp. 199–220Google Scholar
 Gibbons, J., Holm, D.D., Tronci, C.: Vlasov moments, integrable systems and singular solutions. Phys. Lett. A 372(7), 1024–1033 (2008)MathSciNetCrossRefMATHGoogle Scholar
 Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)MathSciNetCrossRefMATHGoogle Scholar
 Guillemin, Victor, Sternberg, Shlomo: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
 Hald, O.H.: Convergence of vortex methods for Euler’s equations. II. SIAM J. Numer. Anal. 16(5), 726–755 (1979)MathSciNetCrossRefMATHGoogle Scholar
 Holm, D.D., Lysenko, W.P., Scovel, J.C.: Moment invariants for the Vlasov equation. J. Math. Phys. 31(7), 1610–1615 (1990)MathSciNetCrossRefMATHGoogle Scholar
 Holm, D.D., Nitsche, M., Putkaradze, V.: Euleralpha and vortex blob regularization of vortex filament and vortex sheet motion. J. Fluid Mech. 555, 149–176 (2006)MathSciNetCrossRefMATHGoogle Scholar
 Hörmander, L.: The analysis of linear partial differential operators. I, Classics in Mathematics, Springer, Berlin, (2003), Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin]Google Scholar
 Jackson, J.D.: Classical electrodynamics, 2nd edn. Wiley, New York (1975)MATHGoogle Scholar
 Kanso, E., Tsang, A.C.H.: Dipole models of selfpropelled bodies. Fluid Dyn. Res. 46(6), 061407 (2014)CrossRefGoogle Scholar
 Le Dizès, S., Verga, A.: Viscous interactions of two corotating vortices before merging. J. Fluid Mech. 467, 389–410 (2002)MathSciNetCrossRefMATHGoogle Scholar
 Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids. Springer, London (1994)CrossRefMATHGoogle Scholar
 Marsden, J.E., Weinstein, A.: Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Phys. D 7(1–3), 305–323 (1983)MathSciNetCrossRefMATHGoogle Scholar
 Melander, M.V., Zabusky, N.J., Mcwilliams, J.C.: Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303–340 (1988)MathSciNetCrossRefMATHGoogle Scholar
 Meunier, P., Le Dizès, S., Leweke, T.: Physics of vortex merging. C. R. Phys. 6(4–5), 431–450 (2005)CrossRefGoogle Scholar
 Mumford, D., Michor, P.W.: On Euler’s equation and ‘EPDiff’. J. Geom. Mech. 5(3), 319–344 (2013)MathSciNetCrossRefMATHGoogle Scholar
 Nicolaides, R.A.: Construction of higher order accurate vortex and particle methods. Appl. Numer. Math. 2(3–5), 313–320 (1986)MathSciNetCrossRefMATHGoogle Scholar
 Nagem, R., Sandri, G., Uminsky, D., Wayne, C.E.: Generalized HelmholtzKirchhoff model for twodimensional distributed vortex motion. SIAM J. Appl. Dyn. Syst 8(1), 160–179 (2009)MathSciNetCrossRefMATHGoogle Scholar
 Oliver, M., Shkoller, S.: The vortex blob method as a secondgrade nonNewtonian fluid. Commun. Partial Differ. Equ. 22, 295–314 (2001)MathSciNetCrossRefMATHGoogle Scholar
 Rosenhead, L.: The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 134(823), 170–192 (1931)CrossRefMATHGoogle Scholar
 Rossi, L.F.: Merging computational elements in vortex simulations. SIAM J. Sci. Comput. 18(4), 1014–1027 (1997)MathSciNetCrossRefMATHGoogle Scholar
 Rossi, L.F.: Achieving highorder convergence rates with deforming basis functions. SIAM J. Sci. Comput. 26(3), 885–906 (2005)MathSciNetCrossRefMATHGoogle Scholar
 Shelley, M.J., Saintillan, D.: Orientational order and instabilities in suspensions of selflocomoting rods. Phys. Rev. Lett. 99 (2007)Google Scholar
 Shelley, M.J., Saintillan, D.: Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations. Phys. Rev. Lett. 100 (2008)Google Scholar
 Smith, S.G.L.: How do singularities move in potential flow? Phys. D 240(20), 1644–1651 (2011)MathSciNetCrossRefMATHGoogle Scholar
 Tchieu, A.A., Kanso, E., Newton, P.K.: The finitedipole dynamical system. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468(2146), 3006–3026 (2012)MathSciNetCrossRefGoogle Scholar
 Tur, A., Yanovsky, V., Kulik, K.: Vortex structures with complex points singularities in twodimensional Euler equations. New exact solutions. Phys. D 240(13), 1069–1079 (2011)MathSciNetCrossRefMATHGoogle Scholar
 Uminsky, D., Wayne, C.E., Barbaro, A.: A multimoment vortex method for 2d viscous fluids. J. Comput. Phys. 231, 1705–1727 (2012)MathSciNetCrossRefMATHGoogle Scholar
 Weiss, J.B., McWilliams, J.C.: Temporal scaling behavior of decaying twodimensional turbulence. Phys. Fluids A Fluid Dyn. 5(3), 608–621 (1993)CrossRefMATHGoogle Scholar
 Yanovsky, V.V., Tur, A.V., Kulik, K.N.: Singularities motion equations in 2dimensional ideal hydrodynamics of incompressible fluid. Phys. Lett. A 373(29), 2484–2487 (2009)CrossRefMATHGoogle Scholar
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