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

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Outline

At micro, energy ∼ norm. How can macro be different from micro?

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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

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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

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Background I

Let u be a 3D vector field on a smoothly bounded domain Ω ∈ R3. Minimization of the “energy” ∥u∥2 yeilds u = 0 (regardless of boundary conditions).

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Background I

Let u be a 3D vector field on a smoothly bounded domain Ω ∈ R3. 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

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Background I

Let u be a 3D vector field on a smoothly bounded domain Ω ∈ R3. 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 flux condition ∫

Σ n · u d2x = Φ are given.

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Background I

Let u be a 3D vector field on a smoothly bounded domain Ω ∈ R3. 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 flux condition ∫

Σ n · u d2x = Φ are given.

Suppose that a helicity ∫

Ω u · (curl−1u)d3x is given. Minimization of

∥u∥2 yields a Beltrami field: ∇ × u = µu.

• Z. Yoshida

topological constraints 2015/08/17 3 / 23

Background I

Let u be a 3D vector field on a smoothly bounded domain Ω ∈ R3. 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 flux condition ∫

Σ n · u d2x = Φ are given.

Suppose that a helicity ∫

Ω u · (curl−1u)d3x is given. Minimization of

∥u∥2 yields a Beltrami field: ∇ × u = µu. 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.

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Background II

What is the helicity ?

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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.).

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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).

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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

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Foliated phase space

Figure: The energy (Hamiltonian) may have a nontrivial distribution on each leaf

• f the foliated phase space.
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Basic Formulation

Hamiltonian system: ∂tu = J∂uH with state vector u ∈ X (phase space), Poisson operator J defining a Poisson bracket {F, G} = ⟨∂uF, J∂uG⟩, and a Hamiltonian H ∈ C ∞

{ , }(X) (Poisson algebra).

The adjoint representation: d dt F = {F, H} = −adHF. Canonical system: J = ( I −I ) → symplectic geometry Noncanonical system has topological defects: Ker(J) = Coker(J). Casimir invariant: J∂uC = 0, i.e., Ker(J) ∋ v = ∂uC.

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The origin of Casimir invariant (a tutorial example) I

Let us start with a 6-dimensional phase space: z := (q1, q2, q3, p1, p2, p3)T ∈ Xz = R6, (1)

• n which we define a canonical Poisson bracket

{F, G}z := (∂zF, Jz∂zG) (2) Jz = Jc := ( I −I ) . (3) Denoting q = (q1, q2, q3)T and p = (p1, p2, p3)T, we define ω := q × p ∈ Xω. (4) We reduce C ∞(Xz) to C ∞(Xω): {F(q × p), G(q × p)}z = {F(ω), G(ω)}ω := (∂ωF, (−ω) × ∂ωG). (5)

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The origin of Casimir invariant (a tutorial example) II

Denoting Jω =   ω3 −ω2 −ω3 ω1 ω2 −ω1   , (6) we may write {F(ω), G(ω)}ω := (∂ωF, Jω∂ωG), (7) which is the so(3) Lie-Poisson bracket. The reduced Poisson algebra, to be denoted by C ∞

{ , }ω(Xω), is

noncanonical, having a Casimir invariant C = 1 2|ω|2. (8) Physically, Xω 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 Xω may be identified with so(3).

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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: ad∗

Cz = Jz∂zC =

( ω × q ω × p ) . (9) The conjugate variable θ such that {θ, C}z = ⟨∂zθ, ad∗

Cz⟩ = 1 is the

longitudinal angle around the axis of ω. Recovering θ, we define a 4D phase space of ζ =

t(ω1, arctan(ω2/ω3), C, θ), on which

Jζ :=   

1 −1 1 −1

   . (10) A Hamiltonian H including θ can unfreeze C.

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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 ∈ XZ be a 2-component field on a base manifold T 2, on which we define a canonical Poisson bracket {F, G}Z := (∂ZF, JZ∂ZG) JZ = Jc := ( I −I ) . (11) We define ω := [Q, P] = dQ ∧ dP. (12) We reduce C ∞(XZ) to C ∞(Xω): {F([Q, P]), G([Q, P])}Z = {F(ω), G(ω)}ω := (∂ωF, [ω, ∂ωG]). (13) We can formulate a 3D compressible system, which, however, involves somewhat nontrivial generalizations.

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Hierarchy of 2D vortex dynamics (1)

Brackets

Table: Hierarchy of two-dimensional vortex systems. Here [Q, P] = ∂yQ∂xP − ∂xQ∂yP.

state Poisson operator Casimir invariants (I) ω [ω, ◦] C0 = ∫ d2x f (ω) (II) ( ω ψ ) ( [ω, ◦] [ψ, ◦] [ψ, ◦] ) C1 = ∫ d2x ωg(ψ) C2 = ∫ d2x f (ψ) (III)   ω ψ ˇ ψ     [ω, ◦] [ψ, ◦] [ ˇ ψ, ◦] [ψ, ◦] [ ˇ ψ, ◦]   C2 = ∫ d2x f (ψ) C3 = ∫ d2x h(ψ ˇ ψ) C4 = ∫ d2x ˇ f ( ˇ ψ)

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Hierarchy of 2D vortex dynamics (2)

Hamiltonians

We denote by ω = −∆ϕ the vorticity with ∆ being the Laplacian and V =

t(∂yϕ, −∂xϕ).

Given a Hamiltonian HE(ω) = −1 2 ∫ d2x ω ∆−1ω, the system (I) is the vorticity equation for Eulerian flow, ∂tω + V · ∇ω = 0. If ψ is the Gauss potential of a magnetic field B =

t(∂yψ, −∂xψ),

and the Hamiltonian is HRMHD(ω, ψ) = −1 2 ∫ d2x [ ω ∆−1ω + ψ ∆ψ ] , the system (II) is the reduced MHD equation, ∂tω + V · ∇ω = J × B, ∂tψ + V · ∇ψ = 0.

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Hierarchy of 2D vortex dynamics (3)

Phantom field

If H is independent of ψ, the dynamics of ω is unaffected by ψ, while both ω and ψ obey the same evolution equation. Then, we call ψ a phantom which can be chosen arbitrarily without changing the dynamics of the actual field ω. At the special choice of ψ = ω, both C1 and C2 evaluate as C0, i.e., C0 is “subsumed” by C1 and C2 as their special value. Indeed, C1 and C2 carry a larger amount of information of the system (I). A modification of the Hamiltonian to involve ψ destroys the constancy of C0; the electromagnetic interaction is a physical example

• f such a modification.
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Hierarchy of 2D vortex dynamics (4)

Integrability of topological constraints

An interesting consequence of extending the system from (I) to (II) is found in the integrability of the Ker(JI). In (I), Ker(JI(ω)) = {ψ; [ω, ψ] = 0}, implying that ψ and ω are related by ψ = η(ζ), ω = ξ(ζ). As far as ξ is a monotonic function, we may write ψ = η(ξ−1(ω)), which we can integrate to obtain C0(ω) with f (ω) such that f ′(ω) = η(ξ−1(ω)). Other elements of Ker(JI(ω)) that are given by nonmonotonic ξ are not integrable to define Casimir invariants. Yet, we can integrate such elements as C1(ω, ψ) in the extended space of (II). In fact, every member of Ker(JI(ω)) can be represented as ∂ωC1 = g(ψ) by choosing ψ in Ker(JI(ω)).

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Topological constraints in ideal MHD

We consider a barotropic MHD: ∂tρ = −∇ · (V ρ), (14) ∂tV = −(∇ × V ) × V − ∇(h + V 2/2) + ρ−1(∇ × B) × B, (15) ∂tB = ∇ × (V × B). (16) The local magnetic flux on an arbitrary co-moving surface σ(t) Φσ(t) = ∫

σ(t)

ν · B d2x =

• ∂σ(t)

τ · A dx is a constant of motion. This is a direct consequence of (16), which implies that the 2-form B is Lie-dragged by the flow V . Because of this infinite set of conservation laws, the magnetic field lines are forbidden to change their topology.

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Hamiltonian structure of magnetohydrodynamics

The phase space X contains the state vector u = (ρ, V , B)T, and the Hamiltonian and Poisson operator are given as follows: H = ∫

Ω

{ ρ [V 2 2 + E(ρ) ] + B2 2 } d3x, (17) J = (

−∇· −∇ −ρ−1(∇ × V )× ρ−1(∇ × ◦) × B ∇ × (

• × ρ−1B

)

) . (18) The Poisson operator J has well-known Casimir invariants: C1 = ∫

Ω

ρ d3x, (19) C2 = 1 2 ∫

Ω

A · B d3x, (20) C3 = ∫

Ω

V · B d3x, (21)

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Extension of the phase space

To formulate the local magnetic flux as a Casimir invariant, we extend the phase space in order to include topological indexes information in the set

• f dynamical variables.

Adding a 2-form ˇ B, which we call a phantom field, to the MHD variables, gives the extended phase space state vector ˜ u = (ρ, V , B, ˇ B)T, (22)

• n which we define a degenerate Poisson manifold by

˜ J =   

−∇· −∇ −ρ−1(∇ × V )× ρ−1(∇ × ◦) × B ρ−1(∇ × ◦) × ˇ B ∇ × (

• × ρ−1B

) ∇ × (

• × ρ−1 ˇ

B )

  . (23) Using the same Hamiltonian (17), we obtain an extended dynamics governed by exactly the same equations (14)-(16) together with an additional equation ∂t ˇ B = ∇ × (V × ˇ B). (24)

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Casimir invariants

The extended Poisson operator (23) has the set of Casimir invariants composed of C1, C2, and a new cross helicity C4 = ∫

Ω

A · ˇ B d3x, (25) as well as a phantom magnetic helicity C5 = 1 2 ∫

Ω

ˇ A · ˇ B d3x. (26) Interestingly, the original (standard) cross helicity C3 = ∫

Ω V · B d3x

is no longer a Casimir invariant of the extended system, although it is still a constant of motion. The constancy of C3 is now due to the “symmetry” of a Hamiltonian with ignorable dependence on the phantom field ˇ B;

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Application to tearing modes (1)

Helical flux Casimir = singular Casimir

The determining equation of the cross helicity: ˜ J ∂˜

uC(B, ˇ

B) = (0, ρ−1[(∇ × ∂BC) × B + (∇ × ∂ ˇ

BC) × ˇ

B], 0, 0)T = 0. Here, we are interested in the singularity at which the rank of ˜ J drops; there is a pair of B∗ = ∇ × A∗ and ˇ B∗ = ∇ × ˇ A∗ such that the two terms on the right-hand side vanish separately, i.e. B∗ × (∇ × ˇ A∗) = (∇ × A∗) × ˇ B∗ = 0. (27) We let C ∗

4 := C4(B∗, ˇ

B∗) and call it a singular cross helicity. A significance of C ∗

4 is (in addition to ˜

J ∂˜

uC ∗ 4 = 0)

J ∂uC ∗

4 = 0.

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Application to tearing modes (2)

Singular Casimir invariant arising from resonant singularity

A trivial solution of (27) is A∗ = ˇ A∗, by which C ∗

4 coincides with C2.

Apparently Rank( ˜ J ) changes at A∗ = ˇ A∗, where the conventional cross helicity C3 resides as a singular Casimir invariants. Interestingly, we may find nontrivial, hyperfunction solutions emerging from the resonance singularity of the differential equation (27). Here we solve (27) for ˇ A∗ by giving B∗. The determining equation can be rewritten as ∇ × ˇ A∗ = ηB∗ (28) with some scalar function η. The singular solution, including “sheet current”, gives “tearing-mode”.

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Application to tearing modes (3)

Tearing mode = equilibrium on a singular Casimir leaf

We consider ∂uHµ = 0, (29) for Hµ = H − µ1C1 − µ2C2 − µ4C4 − µ5C5. We are not demanding ∂˜

uHµ = 0; hence, the solution of (29) is not

necessarily an equilibrium point. However, if we evaluate (29) at the singularity B = B∗ and ˇ B = ˇ B∗, we obtain ∂uHµ|B∗, ˇ

B∗ =

(

V 2/2 + h − µ1 ρV B∗ − µ2A∗ − µ4 ˇ A∗

) = 0. (30) At the solution M∗ of (30), ˜ J ∂˜

uHµ|M∗ = 0, i.e., M∗ is an equilibrium

which bifurcates from the singularity B = B∗ and ˇ B = ˇ B∗. This equilibrium is a “tearing mode” satisfying ∇ × B∗ − µ2B∗ − µ4 ˇ B∗ = 0, (31) where ˇ B∗ is the hyperfunction stemming from the resonant singularity.

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Application to tearing modes (4)

Unfreezing the singular Casimir invariant giving rise to tearing-mode instabilities

We can unfreeze the singular cross helicity C ∗

4 by making a canonical

pair with an angle variable θ. By introducing a perturbed Hamiltonian H(˜ u, θ), C ∗

4 is no longer a

constant.

1 2

tearingmode instability

• B

B B

• †

B1

1

ω B1

• B

Figure: Bifurcation diagram and tearing-mode instability.

ZY & R. L. Dewar; Helical bifurcation and tearing mode in a plasma — a description based on Casimir foliation, J. Phys. A: Math. Theor. 45 (2012), 365502 (36pp).

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Conclusion

1 The example of rigid-body system suggests us to interpret Casimir

invariants as “adiabatic invariants”, frozen by separation of (microscopic) angle variables.

2 Embedding a degenerate Poisson manifold into a higher-dimensional

phase space by adding phantom fields, we can “integrate” the topological constraints (which are not represented by Casimir invariants in the original phase space) as cross helicities.

3 Fluid/plasma systems can be viewed as infinite-dimensional

generalization of the rigid-body system. Infinitely many topological constraints (in principle, Kelvin’s circulation law) can be integrated as cross-helicities.

4 Unfreezing Casimir invariants gives rise to instabilities that are

forbidden in the ideal macroscopic model.

ZY & P. J. Morrison; A hierarchy of noncanonical Hamiltonian systems: circulation laws in an extended phase space, Fluid Dyn. Res. 46 (2014), 031412 1–21; arXiv:1401.7698

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