Topological constraints and structures in macro (fluid and plasma) - PowerPoint PPT Presentation

Topological constraints and structures in macro (fluid and plasma) systems Z. Yoshida University of Tokyo 2015.8.17 Z. Yoshida topological constraints 2015/08/17 1 / 23

Outline At micro, energy ∼ norm. How can macro be different from micro? Z. Yoshida topological constraints 2015/08/17 2 / 23

Outline At micro, energy ∼ norm. How can macro be different from micro? Topological constraints in macro systems → “effective energy” → structure Casimir invariants → foliation of phase space Z. Yoshida topological constraints 2015/08/17 2 / 23

Outline At micro, energy ∼ norm. How can macro be different from micro? Topological constraints in macro systems → “effective energy” → structure Casimir invariants → foliation of phase space Unfreezing Casimir invariants → relaxation Z. Yoshida topological constraints 2015/08/17 2 / 23

Background I Let u be a 3D vector field on a smoothly bounded domain Ω ∈ R 3 . Minimization of the “energy” ∥ u ∥ 2 yeilds u = 0 (regardless of boundary conditions). Z. Yoshida topological constraints 2015/08/17 3 / 23

Background I Let u be a 3D vector field on a smoothly bounded domain Ω ∈ R 3 . Minimization of the “energy” ∥ u ∥ 2 yeilds u = 0 (regardless of boundary conditions). Suppose that u is divergence-free. Minimization of ∥ u ∥ 2 yields Z. Yoshida topological constraints 2015/08/17 3 / 23

Background I Let u be a 3D vector field on a smoothly bounded domain Ω ∈ R 3 . Minimization of the “energy” ∥ u ∥ 2 yeilds u = 0 (regardless of boundary conditions). Suppose that u is divergence-free. Minimization of ∥ u ∥ 2 yields ∇ × u = 0 , which is no longer zero when a boundary condition n · u = g and a Σ n · u d 2 x = Φ are given. ∫ flux condition Z. Yoshida topological constraints 2015/08/17 3 / 23

Background I Let u be a 3D vector field on a smoothly bounded domain Ω ∈ R 3 . Minimization of the “energy” ∥ u ∥ 2 yeilds u = 0 (regardless of boundary conditions). Suppose that u is divergence-free. Minimization of ∥ u ∥ 2 yields ∇ × u = 0 , which is no longer zero when a boundary condition n · u = g and a Σ n · u d 2 x = Φ are given. ∫ flux condition Ω u · ( curl − 1 u ) d 3 x is given. Minimization of ∫ Suppose that a helicity ∥ u ∥ 2 yields a Beltrami field : ∇ × u = µ u . Z. Yoshida topological constraints 2015/08/17 3 / 23

Background I Let u be a 3D vector field on a smoothly bounded domain Ω ∈ R 3 . Minimization of the “energy” ∥ u ∥ 2 yeilds u = 0 (regardless of boundary conditions). Suppose that u is divergence-free. Minimization of ∥ u ∥ 2 yields ∇ × u = 0 , which is no longer zero when a boundary condition n · u = g and a Σ n · u d 2 x = Φ are given. ∫ flux condition Ω u · ( curl − 1 u ) d 3 x is given. Minimization of ∫ Suppose that a helicity ∥ u ∥ 2 yields a Beltrami field : ∇ × u = µ u . The topology of Ω pertains the spectrum of the curl operator.: ZY & Y. Giga, Remarks on spectra of operator rot , Math. Z. 204 (1990), 235-245. Z. Yoshida topological constraints 2015/08/17 3 / 23

Background II What is the helicity ? Z. Yoshida topological constraints 2015/08/17 4 / 23

Background II What is the helicity ? The helicity is a constant of motion of a vortex dynamics system (such as fluid, plasma, (quantum field), etc.). Z. Yoshida topological constraints 2015/08/17 4 / 23

Background II What is the helicity ? The helicity is a constant of motion of a vortex dynamics system (such as fluid, plasma, (quantum field), etc.). The constancy of the helicity is due to a “topological constraint” on the Hamiltonian system; it is not to a “symmetry’ of a specific Hamiltonian (i.e., it conserves independed of the choice of a Hamiltonian). Z. Yoshida topological constraints 2015/08/17 4 / 23

Background II What is the helicity ? The helicity is a constant of motion of a vortex dynamics system (such as fluid, plasma, (quantum field), etc.). The constancy of the helicity is due to a “topological constraint” on the Hamiltonian system; it is not to a “symmetry’ of a specific Hamiltonian (i.e., it conserves independed of the choice of a Hamiltonian). How can the helicity constraint (and other topological constraints) be unfrozen? ZY & P. J. Morrison, Unfreezing Casimir invariants: singular perturbations giving rise to forbidden instabilities, in Nonlinear physical systems: spectral analysis, stability and bifurcation , Ed. by O. N. Kirillov and D. E. Pelinovsky, (ISTE and John Wiley and Sons, 2014) Chap. 18, pp. 401–419. arXiv:1303.0887 Z. Yoshida topological constraints 2015/08/17 4 / 23

Foliated phase space Figure: The energy (Hamiltonian) may have a nontrivial distribution on each leaf of the foliated phase space. Z. Yoshida topological constraints 2015/08/17 5 / 23

Basic Formulation Hamiltonian system: ∂ t u = J ∂ u H with state vector u ∈ X (phase space), Poisson operator J defining a Poisson bracket { F , G } = ⟨ ∂ u F , J ∂ u G ⟩ , and a Hamiltonian H ∈ C ∞ { , } ( X ) (Poisson algebra). The adjoint representation: d dt F = { F , H } = − ad H F . ( ) 0 I Canonical system: J = → symplectic geometry − I 0 Noncanonical system has topological defects: Ker ( J ) = Coker ( J ). Casimir invariant: J ∂ u C = 0, i.e., Ker ( J ) ∋ v = ∂ u C . Z. Yoshida topological constraints 2015/08/17 6 / 23

The origin of Casimir invariant (a tutorial example) I Let us start with a 6-dimensional phase space: z := ( q 1 , q 2 , q 3 , p 1 , p 2 , p 3 ) T ∈ X z = R 6 , (1) on which we define a canonical Poisson bracket { F , G } z := ( ∂ z F , J z ∂ z G ) (2) ( ) 0 I J z = J c := . (3) − I 0 Denoting q = ( q 1 , q 2 , q 3 ) T and p = ( p 1 , p 2 , p 3 ) T , we define ω := q × p ∈ X ω . (4) We reduce C ∞ ( X z ) to C ∞ ( X ω ): { F ( q × p ) , G ( q × p ) } z = { F ( ω ) , G ( ω ) } ω := ( ∂ ω F , ( − ω ) × ∂ ω G ) . (5) Z. Yoshida topological constraints 2015/08/17 7 / 23

The origin of Casimir invariant (a tutorial example) II Denoting   0 ω 3 − ω 2  , J ω = − ω 3 0 ω 1 (6)  ω 2 − ω 1 0 we may write { F ( ω ) , G ( ω ) } ω := ( ∂ ω F , J ω ∂ ω G ) , (7) which is the so (3) Lie-Poisson bracket. The reduced Poisson algebra, to be denoted by C ∞ { , } ω ( X ω ), is noncanonical, having a Casimir invariant C = 1 2 | ω | 2 . (8) Physically, X ω is the phase space of a rigid-body on an inertia frame co-moving with the center of mass. The mechanical degree of freedom is, then, only the angular momentum ω ; the phase space X ω may be identified with so (3). Z. Yoshida topological constraints 2015/08/17 8 / 23

The origin of Casimir invariant (a tutorial example) III In the 6D canonical phase space, C is a Noether charge corresponding to the gauge symmetry of the parameterization ω = q × p : ( ω × q ) ad ∗ C z = J z ∂ z C = . (9) ω × p The conjugate variable θ such that { θ, C } z = ⟨ ∂ z θ, ad ∗ C z ⟩ = 1 is the longitudinal angle around the axis of ω . Recovering θ , we define a 4D phase space of t ( ω 1 , arctan( ω 2 /ω 3 ) , C , θ ), on which ζ =  0 1 0 0  − 1 0 0 0 J ζ :=  . (10)   0 0 0 1  0 0 − 1 0 A Hamiltonian H including θ can unfreeze C . Z. Yoshida topological constraints 2015/08/17 9 / 23

Vortex dynamics Vortex dynamics is as an infinite-dimensional generalization of the aforementioned so (3) noncanonical Lie-Poisson system. Let Z = ( Q ( x ) , P ( x )) T ∈ X Z be a 2-component field on a base manifold T 2 , on which we define a canonical Poisson bracket ( ) 0 I { F , G } Z := ( ∂ Z F , J Z ∂ Z G ) J Z = J c := . (11) − I 0 We define ω := [ Q , P ] = d Q ∧ d P . (12) We reduce C ∞ ( X Z ) to C ∞ ( X ω ): { F ([ Q , P ]) , G ([ Q , P ]) } Z = { F ( ω ) , G ( ω ) } ω := ( ∂ ω F , [ ω, ∂ ω G ]) . (13) We can formulate a 3D compressible system, which, however, involves somewhat nontrivial generalizations. Z. Yoshida topological constraints 2015/08/17 10 / 23

Hierarchy of 2D vortex dynamics (1) Brackets Table: Hierarchy of two-dimensional vortex systems. Here [ Q , P ] = ∂ y Q ∂ x P − ∂ x Q ∂ y P . state Poisson operator Casimir invariants d 2 x f ( ω ) ∫ (I) ω [ ω, ◦ ] C 0 = ( ω ( [ ω, ◦ ] d 2 x ω g ( ψ ) ) ) ∫ [ ψ, ◦ ] C 1 = (II) ∫ d 2 x f ( ψ ) ψ [ ψ, ◦ ] 0 C 2 = [ ˇ ∫ d 2 x f ( ψ )  ω   [ ω, ◦ ] [ ψ, ◦ ] ψ, ◦ ]  C 2 = d 2 x h ( ψ ˇ ∫ (III) ψ [ ψ, ◦ ] 0 0 C 3 = ψ )     ˇ [ ˇ d 2 x ˇ f ( ˇ ∫ ψ ψ, ◦ ] 0 0 C 4 = ψ ) Z. Yoshida topological constraints 2015/08/17 11 / 23

Hierarchy of 2D vortex dynamics (2) Hamiltonians We denote by ω = − ∆ ϕ the vorticity with ∆ being the Laplacian and t ( ∂ y ϕ, − ∂ x ϕ ). V = Given a Hamiltonian H E ( ω ) = − 1 ∫ d 2 x ω ∆ − 1 ω, 2 the system (I) is the vorticity equation for Eulerian flow, ∂ t ω + V · ∇ ω = 0 . t ( ∂ y ψ, − ∂ x ψ ), If ψ is the Gauss potential of a magnetic field B = and the Hamiltonian is H RMHD ( ω, ψ ) = − 1 ∫ d 2 x ω ∆ − 1 ω + ψ ∆ ψ [ ] , 2 the system (II) is the reduced MHD equation, ∂ t ω + V · ∇ ω = J × B , ∂ t ψ + V · ∇ ψ = 0 . Z. Yoshida topological constraints 2015/08/17 12 / 23