Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics

J Nonlinear Sci DOI 10.1007/s00332-017-9367-4 Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics Darryl D. Holm1 · Henry O. Jacobs1 Received: 10 November 2015 / Accepted: 18 December 2016 © The Author(s) 2017. This article is published with open access at Springerlink.com 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 dynamics Mathematics Subject Classification 76M23 · 76M60 · 70H15 Communicated by Paul Newton. B Darryl D. Holm 1 Department of Mathematics, Imperial College, London SW7 2AZ, UK 123 J Nonlinear Sci 1 Introduction Vortices are important in hydrodynamics because they are the sources for the incompressible flow field. The vorticity distribution at any instant of time determines both the current state of the flow and its future evolution, for given boundary conditions. This property holds for any Hamiltonian system, and it can indeed be shown that the dynamics of vortices can be usefully expressed in Hamiltonian form. In the vorticity and stream function formulation of an ideal incompressible planar fluid, the evolution of the vorticity distribution ω(x, y, t) is given by ∂t ω − {ω, ψ} ≡ ∂t ω − ∂x ω ∂ y ψ + ∂ y ω ∂x ψ = 0 , (1) where ω = −ψ is the vorticity, ψ is the stream function, and  = ∂x x + ∂ yy is the Laplace operator. The corresponding (x, y) components of the Eulerian velocity field are given by (u, v) = (∂ y ψ, −∂x ψ). If one is willing to view the vorticity ω as a distribution, one can consider point vortex solutions. In particular, point vortices are obtained if one considers the vorticity solution ansatz  ω(z, t) = i (t)δzi (t) , i where i (t) ∈ R, z = (x, y) ∈ R2 and δzi (t) is the Dirac delta distribution centered at the point z i (t) = (xi (t), yi (t)) ∈ R2 at a given time t ∈ R. Substitution of this ansatz into (1) yields the following well-known finite dimensional system in the form of Hamilton’s canonical equations,  di = 0, ψ(z, t) = i (t)G(z − z i (t)), dt i dxi = ∂ y ψ(z i ), dt (2) dyi = −∂x ψ(z i ), dt where G(z) = −(2π )−1 ln(z) is the Green’s function for the planar Laplacian. 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). For example, one may consider the regularized form of the vorticity equation given by choosing a translationally and rotationally invariant smoothing kernel K δ of width δ > 0 and defining the regularized vorticity as K δ ∗ ω = −ψ while continuing to use (1) to evolve ω in time. For example, K δ (z) = exp(−z2 /δ 2 ) is considered in Beale and Majda (1985). In this case, the point vortex ansatz yields (2) again, except that the singular Green’s function G is replaced by the smooth kernel G δ (z) := K δ ∗ G(z) = 123  1  Ei(−z2 /δ 2 ) − 2 ln(z) , 4π (3) J Nonlinear Sci where Ei(·) denotes the exponential integral function. The vorticity kernel G δ has no singularity at the origin for δ > 0 and is known as a vortex blob. This system is the starting point for the vortex blob method, introduced in Chorin (1973) (albeit with a different regularization). The economy of the vortex blob method derives from the property that Dirac delta distributions are hyper-local (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, ∂x δzi ). In this paper, we study the more general vorticity solution ansatz,  imn (t)∂xm ∂ yn δzi . ω(z, t) = i,m,n We find that this ansatz yields a closed finite dimensional system which generalizes vortex blobs. We call these new carriers of vorticity multipole vortex blobs or MVBs. 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 123 J Nonlinear Sci 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 t (...truncated)