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

Rigidity of MHD equilibrium states to smooth ideal motion

David Pfefferl´ e1 Lyle Noakes1 Yao Zhou2

1The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia 2Princeton 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 diffeomorphisms1[Arnold, 1966]

ϕt :M ⊆ R3 → M with ϕ0 = id x0 → x(t) = ϕt(x0)

• Eulerian velocity field is represented by the smooth time-variation

ut := ∂tϕt ◦ ϕ−1

t

∈ X(M) i.e. u(x(t), t) = ∂tϕt(x0) = ˙ x(t)

x0

x(t) = ϕt(x0)

• u(x(t), t)

ϕt

ϕ−1

t

1smooth map with smooth inverse D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 2 / 17

Frozen-in condition as advection of flux

Let At be the vector potential (one-form). The frozen-in condition is At := ϕ−1

t ∗A0,

i.e. A(x(t), t) · w = A(x0, 0) · v

∀v ∈ R3, w = Jv where J = dϕtx0 is the Jacobian matrix.

• v
• A(x0, 0)
• w
• A(x(t), t)

x0 x(t)

equivalently ∂tAt + £utAt = 0 i.e. Ohm’s law E + u × B = 0

where2 B = ∇ × A and E = −∂tA − ∇(A · u).

2The magnetic field is defined via dAt = iBtω 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 group3 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 ∗

l(u, (A, ρ)) :=

• M

1 2|u|2ρ − 1 2|B|2ω

where ω = dx1 ∧ dx2 ∧ dx3 is the natural volume form on R3

• Use as a right-invariant Lagrangian on TG

L(ϕt, ∂tϕt, αt, ∂tαt) := l(∂tϕt ◦ ϕ−1

t , ϕ−1 t ∗∂tαt)

3Here V ∗ = Ω1(M) × Ω3(M) D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 4 / 17

• Hold ∂tαt := a0 fixed, i.e. impose advection laws

At = ϕ−1

t ∗A0 ⇐

⇒ (∂t + £ut)At = 0 ⇐ ⇒ E + u × B = 0 ρt = ϕ−1

t ∗ρ0 ⇐

⇒ (∂t + £ut)ρt = 0 ⇐ ⇒ ∂tρ + ∇ · (ρu) = 0

• Restrict to La0(ϕt, ∂tϕt) := l(∂tϕt ◦ ϕ−1

t , ϕ−1∗a0)

Theorem [Holm et al., 1998; Marsden et al., 1984] The variational principle on Diff(M) of Sa0[ϕ] =

t2

• t1

La0(ϕt, ∂tϕt)dt is equivalent to the constrained variational problem on Diff(M) × V ∗ of S[u, A, ρ] =

t2

• t1

l(u, (A, ρ))dt subject to δu = (∂t + £u)η δA = −£ηA δρ = −£ηρ

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 = ρ0 through

Lagrange multiplier l(u, (A, ρ)) :=

• M

1 2|u|2ρ − 1 2|B|2ω − P(ρ − ρ0)

G0 = SDiff(M) × Ω1(M), g0 = {u ∈ X(M)|∇ · (ρ0u) = 0} × Ω1(M) Incompressible ideal MHD ρ0(∂tu + u · ∇u) + ∇P = J × B ∂tB + ∇ × (u × B) = 0

where “pressure” P is enforcing ∇ · (ρ0u) = 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 =

• M

A · B ω, C =

• M

u · B ω, S =

• M

f(ρ) ω

• invariants define level sets on which the dynamics take place in G
• dynamical problem ⇒ equilibrium problem (Wick rotation)

δS = δT − δV = 0 ⇒ δE = δT + δV = 0 MHD equilibrium ≡ critical points of “effective potential” along the invariant level sets E =

• M

1 2B2ω + Pρ + λA · Bω E.L

⇒ J × B = ∇P

D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 7 / 17

Killing symmetry ⇒ Grad-Shafranov

• Killing vector field Z preserves the metric

£ZX, Y = £ZX, Y + X, £ZY , X, Y ∈ Γ(TM)

• Symmetry is the assumption £ZA = 0 ⇒ £ZB = 0 ⇒ £ZJ = 0
• coordinate-free Grad-Shafranov equation

J × B = ∇P ⇒ Z2δ(Z−2dΨ) + F(µ + F ′) + Z2P ′ = 0

where Ψ = −A(Z), F = B(Z), µ = Z · (∇ × Z)/Z2 “twistness” of Killing field, F ′ = dF/dΨ, P ′ = dP/dΨ and Z2 = Z, Z

• in R3, Killing field = 3 translations + 3 rotations

∆Ψ = V ′(Ψ) R2∇ · ∇Ψ R2

• = −FF ′ − R2P ′

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

• B
• Bh = ∇Ψh × ˆ

z = xˆ y

x y

D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 10 / 17

Hahm-Kulsrud-Taylor problem

• Cartesian “poloidal” plane M ∼

= R × S1

• Grad-Shafranov equation ∆Ψ = V ′(Ψ) can be written

d(∆ΨdΨ) = 0

• initial configuration with line of critical points

Ψ0(x, y) = 1

2x2

B0 = ∇Ψ0 × ˆ z = −xˆ y J0 = 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 ϕǫ and

frozen-in condition Ψǫ := ϕ−1

ǫ ∗Ψ0 = Ψ0 ◦ ϕ−1 ǫ

• seek those that retain force-balance d(∆ΨǫdΨǫ) = 0, ∀ǫ
• let Xǫ = ∂ǫϕǫ ◦ ϕ−1

ǫ

∈ X(M) be the corresponding smooth vector field

• volume-preserving ⇐

⇒ Xǫ divergence-free ⇐ ⇒ Xǫ = ∇ × (Sǫˆ z), ∂ǫΨǫ = Bǫ · ∇Sǫ

where Sǫ(X, Y ) are smooth functions.

• Can we preserve parity of Ψǫ with boundary condition :

Ψǫ(±1, Y ) = Ψ0(±1)(1 + ǫ cos Y ) ? No !

D.Pfefferl´ e (UWA) Rigidity of ideal MHD Simons Meeting 12 / 17

• let Φ := ∂ǫΨǫ
• ǫ=0, with Φ(±1, y) = 1

2 cos y

• differentiate the force-balance condition with respect to ǫ at ǫ = 0

d(∆ΦdΨ0) + d(✟✟

✟

∆Ψ0dΦ) = 0 ⇐ ⇒ d(∆Φdx2) = 0 ⇐ ⇒ ∆∂yΦ = 0 The ǫ-derivative is harmonic (plus a function of x) Φ(x, y) = A cosh x cos y + f(x)

where A =

1 2 cosh 1 = e/(1 + e2).

• at the same time, however,

∂ǫΨǫ

• ǫ=0 = B0 · ∇S0

⇒ Φ = −x∂yS0 i.e. the potential function S0(x, y) is singular at x = 0 S0 = −Acosh x x sin y

• 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

Ψǫ = Ψ0 + ǫΦ + O(ǫ2) x = x0 + ǫX(x0) 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

Ψǫ = Ψ0 + ǫΦ + O(ǫ2) x = x0 + ǫX(x0) 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