Rigidity of MHD equilibrium states to smooth ideal motion e 1 Lyle - PowerPoint PPT Presentation

Rigidity of MHD equilibrium states to smooth ideal motion e 1 Lyle Noakes 1 Yao Zhou 2 David Pfefferl´ 1 The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia 2 Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA Simons Collaboration on Hidden Symmetries and Fusion Energy Meeting March 28-29, 2018 - New York, US

Fluid motion via smooth flow maps • fluid motion is represented by family of diffeomorphisms 1 [Arnold, 1966] ϕ t : M ⊆ R 3 → M with ϕ 0 = id x 0 �→ x ( t ) = ϕ t ( x 0 ) • Eulerian velocity field is represented by the smooth time-variation u t := ∂ t ϕ t ◦ ϕ − 1 ∈ X ( M ) i.e. u ( x ( t ) , t ) = ∂ t ϕ t ( x 0 ) = ˙ x ( t ) t x ( t ) = ϕ t ( x 0 ) x 0 ϕ t u ( x ( t ) , t ) � ϕ − 1 t 1 smooth map with smooth inverse D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 2 / 17

Frozen-in condition as advection of flux Let A t be the vector potential (one-form). The frozen-in condition is ∗ A 0 , A t := ϕ − 1 i.e. A ( x ( t ) , t ) · w = A ( x 0 , 0) · v t ∀ v ∈ R 3 , w = J v where J = dϕ tx 0 is the Jacobian matrix. � A ( x 0 , 0) � A ( x ( t ) , t ) x ( t ) x 0 w � � v equivalently ∂ t A t + £ u t A t = 0 i.e. Ohm’s law E + u × B = 0 where 2 B = ∇ × A and E = − ∂ t A − ∇ ( A · u ) . 2 The magnetic field is defined via dA t = i B t ω where ω is the natural volume-form D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 3 / 17

Lagrangian reduction [Holm et al., 1998; Ono, 1995; Hattori, 1994] • Fluid state is a point in the Lie group 3 G = Diff ( M ) × V ∗ with multiplication ( ϕ, α )( ψ, β ) = ( ϕ ◦ ψ, ψ ∗ α + β ) • Define the lagrangian on the Lie algebra g = X ( M ) × V ∗ � 1 2 | u | 2 ρ − 1 2 | B | 2 ω l ( u , ( A , ρ )) := M where ω = dx 1 ∧ dx 2 ∧ dx 3 is the natural volume form on R 3 • Use as a right-invariant Lagrangian on TG ∗ ∂ t α t ) L ( ϕ t , ∂ t ϕ t , α t , ∂ t α t ) := l ( ∂ t ϕ t ◦ ϕ − 1 t , ϕ − 1 t 3 Here V ∗ = Ω 1 ( M ) × Ω 3 ( M ) D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 4 / 17

• Hold ∂ t α t := a 0 fixed, i.e. impose advection laws ∗ A 0 ⇐ A t = ϕ − 1 ⇒ ( ∂ t + £ u t ) A t = 0 ⇐ ⇒ E + u × B = 0 t ∗ ρ 0 ⇐ ρ t = ϕ − 1 ⇒ ( ∂ t + £ u t ) ρ t = 0 ⇐ ⇒ ∂ t ρ + ∇ · ( ρ u ) = 0 t t , ϕ − 1 ∗ a 0 ) • Restrict to L a 0 ( ϕ t , ∂ t ϕ t ) := l ( ∂ t ϕ t ◦ ϕ − 1 Theorem [Holm et al., 1998; Marsden et al., 1984] The variational principle on Diff ( M ) of t 2 � S a 0 [ ϕ ] = L a 0 ( ϕ t , ∂ t ϕ t ) dt t 1 is equivalent to the constrained variational problem on Diff ( M ) × V ∗ of t 2 � S [ u , A , ρ ] = l ( u , ( A , ρ )) dt subject to t 1 δu = ( ∂ t + £ u ) η δA = − £ η A δρ = − £ η ρ D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 5 / 17

Incompressible ideal MHD equations as action-minimising curves on Diff ( M ) × V ∗ • density-preserving diffeomorphisms by imposing ρ = ρ 0 through Lagrange multiplier � 2 | u | 2 ρ − 1 2 | B | 2 ω − P ( ρ − ρ 0 ) 1 l ( u , ( A , ρ )) := M G 0 = S Diff ( M ) × Ω 1 ( M ) , g 0 = { u ∈ X ( M ) |∇ · ( ρ 0 u ) = 0 } × Ω 1 ( M ) Incompressible ideal MHD ρ 0 ( ∂ t u + u · ∇ u ) + ∇ P = J × B ∂ t B + ∇ × ( u × B ) = 0 where “pressure” P is enforcing ∇ · ( ρ 0 u ) = 0 on a simply connected domain, J = ∇ × B D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 6 / 17

The MHD equilibrium problem • MHD has several invariants, e.g. � � � H = A · B ω, C = u · B ω, S = f ( ρ ) ω M M M • invariants define level sets on which the dynamics take place in G • dynamical problem ⇒ equilibrium problem (Wick rotation) δ S = δ T − δ V = 0 ⇒ δ E = δ T + δ V = 0 MHD equilibrium ≡ critical points of “effective potential” along the invariant level sets � E.L 1 2 B 2 ω + Pρ + λ A · B ω E = ⇒ J × B = ∇ P M D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 7 / 17

Killing symmetry ⇒ Grad-Shafranov • Killing vector field Z preserves the metric £ Z � X, Y � = � £ Z X, Y � + � X, £ Z Y � , X, Y ∈ Γ( TM ) • Symmetry is the assumption £ Z A = 0 ⇒ £ Z B = 0 ⇒ £ Z J = 0 • coordinate-free Grad-Shafranov equation Z 2 δ ( Z − 2 d Ψ) + F ( µ + F ′ ) + Z 2 P ′ = 0 J × B = ∇ P ⇒ where Ψ = − A ( Z ) , F = B ( Z ) , µ = Z · ( ∇ × Z ) /Z 2 “twistness” of Killing field, F ′ = dF/d Ψ , P ′ = dP/d Ψ and Z 2 = � Z, Z � • in R 3 , Killing field = 3 translations + 3 rotations � ∇ Ψ � = − FF ′ − R 2 P ′ R 2 ∇ · ∆Ψ = V ′ (Ψ) R 2 D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 8 / 17

3D MHD equilibrium • 3D equilibrium without Killing symmetry is an open question • existence of 3D flux-surfaces is weakened by “rational surfaces” D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 9 / 17

Helical field near rational surfaces x y � B � B h = ∇ Ψ h × ˆ z = x ˆ y D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 10 / 17

Hahm-Kulsrud-Taylor problem • Cartesian “poloidal” plane M ∼ = R × S 1 • Grad-Shafranov equation ∆Ψ = V ′ (Ψ) can be written d (∆Ψ d Ψ) = 0 • initial configuration with line of critical points 2 x 2 Ψ 0 ( x, y ) = 1 B 0 = ∇ Ψ 0 × ˆ z = − x ˆ y J 0 = 1 D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 11 / 17

Sequence of equilibria via incompressible ideal motion • Generate equilibria via volume-preserving diffeomorphisms ϕ ǫ and ∗ Ψ 0 = Ψ 0 ◦ ϕ − 1 frozen-in condition Ψ ǫ := ϕ − 1 ǫ ǫ • seek those that retain force-balance d (∆Ψ ǫ d Ψ ǫ ) = 0 , ∀ ǫ • let X ǫ = ∂ ǫ ϕ ǫ ◦ ϕ − 1 ∈ X ( M ) be the corresponding smooth vector field ǫ • volume-preserving ⇐ ⇒ X ǫ divergence-free ⇐ ⇒ X ǫ = ∇ × ( S ǫ ˆ z ) , ∂ ǫ Ψ ǫ = B ǫ · ∇ S ǫ where S ǫ ( X, Y ) are smooth functions. • Can we preserve parity of Ψ ǫ with boundary condition : Ψ ǫ ( ± 1 , Y ) = Ψ 0 ( ± 1)(1 + ǫ cos Y ) ? No ! D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 12 / 17

• let Φ := ∂ ǫ Ψ ǫ � ǫ =0 , with Φ( ± 1 , y ) = 1 2 cos y � • differentiate the force-balance condition with respect to ǫ at ǫ = 0 ✟ ⇒ d (∆Φ dx 2 ) = 0 ⇐ d (∆Φ d Ψ 0 ) + d ( ✟✟ ∆Ψ 0 d Φ) = 0 ⇐ ⇒ ∆ ∂ y Φ = 0 The ǫ -derivative is harmonic (plus a function of x ) Φ( x, y ) = A cosh x cos y + f ( x ) 2 cosh 1 = e/ (1 + e 2 ) . 1 where A = • at the same time, however, � ∂ ǫ Ψ ǫ ǫ =0 = B 0 · ∇ S 0 ⇒ Φ = − x∂ y S 0 � � i.e. the potential function S 0 ( x, y ) is singular at x = 0 S 0 = − A cosh x sin y x • the motion is not supported ⇒ rigidity of MHD equilibrium D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 13 / 17

Singular vector field ⇒ non-ideal tearing Island width scales like √ ǫ Ψ ǫ = Ψ 0 + ǫ Φ + O ( ǫ 2 ) x = x 0 + ǫ X ( x 0 ) ǫ = 0 . 01 D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 14 / 17

Singular vector field ⇒ non-ideal tearing Island width scales like √ ǫ Ψ ǫ = Ψ 0 + ǫ Φ + O ( ǫ 2 ) x = x 0 + ǫ X ( x 0 ) ǫ = 0 . 05 D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 15 / 17

Conclusions • there are no smooth solutions to HKT problem that retain parity of flux-function Ψ( − x, y ) = Ψ( x, y ) • no smooth isotopy between equilibrium states ⇒ dynamically inaccessible via ideal motion • interpretations/workaround • finite resistivity ⇒ tearing layer, etc. . . • avoid critical point by discontinuous magnetic field (current sheet, jump in rotational transform) • force-balance condition relaxed near resonant layer • MHD equilibrium with nested flux-surfaces are extremely rare/exceptional (fine-tuning) D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 16 / 17

Bibliography I V. Arnold, Annales de l’Institut Fourier 16 , 319 (1966). D. D. Holm, J. E. Marsden, and T. S. Ratiu, Advances in Mathematics 137 , 1 (1998), ISSN 0001-8708. T. Ono, Physica D: Nonlinear Phenomena 81 , 207 (1995), ISSN 0167-2789. Y. Hattori, Journal of Physics A: Mathematical and General 27 , L21 (1994). J. E. Marsden, T. Ratiu, and A. Weinstein, Transactions of the American Mathematical Society 281 , 147 (1984), ISSN 00029947. D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 17 / 17