Topological Methods in Fluid Dynamics

Transcription

1 Topological Methods in Fluid Dynamics Gunnar Hornig Topologische Fluiddynamik Ruhr-Universität-Bochum IBZ, Februar 2002 Page 1 of 36 Collaborators: H. v. Bodecker, J. Kleimann, C. Mayer, E. Tassi, S.V. Titov

2 1. Why are we interested in topological features of fluids? Topological properties do not depend on a (continuous/differentiable) deformation of the object. Topological properties are an appropriate description for fluids. Topological properties are often very robust Changes of topological properties are very violent. But where and when topological properties change most often depends on nontopological properties. A topological description requires: Characterisation of the topological properties Description under which conditions they are conserved or destroyed Page 2 of 36

3 2. Hydrodynamics: t ρ + (ρv) = 0 ρ t v + ρv v = p + ν v ν = const., ( v = 0) Navier-Stokes Eq. Equation for the vorticity w = v: t w (v w) = ( 1/ρ p v 2 /2 + 1/ρν v ) t w (v w) = 0 Conservation of vorticity (Kelvin s Theorem) : Approximation of large Reynolds Number: R e := V 0 L 0 /ν and isentropic flow (p = p(ρ)) Page 3 of 36

4 Magnetohydrodynamics: t ρ + (ρv) = 0 ρ t v + ρv v = p + J B E + v B = ηj + other terms ( ) E + t B = 0 B = 0 B = µj Ohm s law ( ) Induction equation: t B (v B) = (ηj + other terms) t B (v B) = 0 Conservation of magnetic flux (Alfvéns s Theorem) : Approximation of large magnetic Reynolds Number: R M := V 0 L 0 /η Page 4 of 36

5 3. Topological Conservation Laws 3.1. Flux conservation t B (v B) = 0 B da = const. for a comoving surface C 2, C 2 (t) Conservation of flux Conservation of field lines Conservation of null points Conservation of knots and linkages of field lines Transport of fields (ω k ) in the flow v(x, t) implies the conservation of an integral over a k-dimensional comoving volume C k. t ω k + L v ω k = 0 ω k = const. F (C k ) Page 5 of 36

6 3.2. For physical space (IR 3 ): t ω 0 α + L v ω 0 α = 0 t α + v α = 0 α(f (x, t), t) = const. Example: α decribes the color in a mixture of paints t ωa 1 + L v ωa 1 = 0 t A + (v A) v A = 0 A dl = const. C 1 (t) Example: A = α t ωb 2 + L v ωb 2 = 0 t B + v B B v + B v = 0 B da = const., C 2 (t) Example: B vorticity in hydrodynamics (Kelvins Theorem) t ωρ 3 + L v ωρ 3 = 0 t ρ + (v ρ) = 0 ρ d 3 x = const. C 3 (t) Example: ρ mass-density in hydrodynamics Page 6 of 36

7 3.3. Conservation laws in space-time L V ωα 0 = 0 V t t α + V α = 0 ωα 0 = α = const. C 0 L V ωa 1 = 0 L V ωab 2 = 0 L V ωa 3 = 0 { t (V t A t ) + V A t A t V = 0 V t t A + (V A) V A A t V t = 0 ωa 1 = A t dt A dl = const. C 1 C 1 C 1 { t (V t A) + (V A) V A t V B = 0 V t t B (V B) + V B + V t A = 0 ωab 2 = B da A dl dt = const. C 2 C 2 C 2 { V t t A t + (VA t ) A V t = 0 t (V t A) (V A) + V A A t t V = 0 ωa 3 = A t dv A da dt = const. C 3 C 3 C 3 L V ωρ 4 = 0 t (ρv t ) + (ρv) = 0 ωρ 4 = ρ dv (4) = const. C 4 C 4 Page 7 of 36

8 3.4. Conservation in IR n s ωα 0 + L V ωα 0 = 0 s α + V α = 0 ωα 0 = α = const. C 0.. s ωρ n + L V ωρ n = 0 s ρ + (ρv) = 0 ωρ 4 = C n ρ dv (n) = const. C n Example: V Hamilton flow on IR (2m), ω (2k) = dq 1 dp 1... dq k dp k yields Poincaré integral invariants Page 8 of 36

9 4. Structure of magnetic fields Long range forces: Gravitation Electromagnetism Electric fields (shielded) Magnetic fields Magnetic fields are found in many astrophysical objects: planets, stars, white dwarfs, pulsars, interstellar clouds, galaxy, intergalactic field Are these all dipolar fields? Page 9 of 36

10 The magnetic field of the Earth s core according to a numerical simulation by G. Glazmeier. Page 10 of 36

11 X-ray image of the Sun, Yohkoh satellite Page 11 of 36

12 Page 12 of 36

13 Even simple magnetic configurations contain linked and knotted field lines. A field line forming a simple torus-knot. A field line linking the central field line of the torus. Page 13 of 36 Can we quantify the knottedness or linkage of flux in a magnetic field?

14 Total The (total) magnetic helicity is defined as H(B) := A }{{ B } d 3 x for B n V = 0, V hel. density where A is the vector potential for the magnetic field B, which is tangent to the boundary V. In terms of the magnetic field only: H(B) = 1 4π B(x ) x x x x 3 B(x) d3 x d 3 x which shows that the helicity is of 2nd order in the magnetic field. The total helicity measures the mutual linkage of flux in the volume. Page 14 of 36

15 5.2. Cross The helicity integral can be considered as a special case of the mutual or cross helicity integral: H(B 1, B 2 ) := A 1 B 2 d 3 x = A 2 B 1 d 3 x for B 1 n V = B 2 n V = 0 V For B = B 1 + B 2 we have the relation V H(B 1 + B 2 ) = }{{} H(B 1 ) + H(B 2 ) }{{} +2 H(B 1, B 2 ) }{{} total helicty self helicity of comp. cross helicity Page 15 of 36

16 5.3. Evolution of helicity The homogeneous Maxwell s equations yield a balance equation for the helicity density: Remarks: t A B }{{} hel. density + (ΦB + E A) = }{{} 2 E B }{{} hel. current hel. source There is no freedom to add certain terms either to the current or to the source since the covariant formulation uniquely determines the helicity current. The helicity density and the helicity current are not gauge invariant, but the source term is gauge invariant. Integrating over a volume yields an expression for the evolution of the total helicity d A B d 3 x = 2 E B d 3 x, dt V V Page 16 of 36

17 5.4. Topology and free energy of a magnetic field Energy of a magnetic field: E m (B) := 1 8π B 2 d 3 x Free energy of a magnetic field: E F := E m (B) E m (B 0 ) where B 0 is the vacuum field satisfying the same boundary conditions. Minimum free energy of a magnetic field: min(e F ) := E m (B ) E m (B 0 ) where B is the lowest energy state accessible from B by an ideal relaxation. Page 17 of 36

18 Every kind of linkage or knottedness contributes to a lower bound for the free energy of a magnetic field. E.g. the total magnetic helicity yields the inequality E F min(e F (B)) C H(B) ([Arnold 1986], [Freedman 1988], [Freedman & He 1991a], [Freedman & He 1991b], [Berger 1993]) There is a need for higher order invariants and new energy limits which measure more complex linkage or knottedness. Page 18 of 36

19 6. t B (v B) = N (1) N : Non-ideal term, e.g. N = ηj If B N = 0 we can rewrite (1) t B (w B) = 0 with w := v v = v B N B 2 Existence of a smooth w slippage solution (See example of J. Kleimann) Existence of a smooth w with exception of points where B = 0 but N 0 Reconnection-like processes B has a X-line classical (2d) reconnection B has a 3d null point null-point reconnection? B N 0 no w exists. Localized N 3d reconnection N 0 globally global dissipation (very slow for astrophysical plasmas) Page 19 of 36

20 A simple reconnection process Movie An evolution of field lines integrated from two cross-sections (black) comoving with the fluid is shown. The cross-section are located outside the non-ideal region (N 0 only in a neighborhood of the z-axis) and initially belong to the same magnetic flux tube. The flux tube is transported with velocity w, which outside the non-ideal region coincides with the plasma velocity v. Page 20 of 36

21 The reconnection occurs along a line, the reconnection line. This is the line along which the virtual transport velocity w is singular (here the z-axis of the box). Page 21 of 36

22 D Reconnection B 0 No flux conserving flow w exists! Movie An evolution of field lines integrated from two cross-sections (black) comoving with the fluid is shown. The cross-section are located outside the non-ideal region (N 0 only in a neighborhood of the z-axis) and initially belong to the same magnetic flux tube. There exists no transport velocity w as in the 2-d case. Thus the flux tube splits as soon as it enters the non-ideal region. The strands of the flux tube flip Page 22 of 36

23 around each other and finally merge again. Note that there exist a covariant formulation where this type of reconnection is defined by a singularity of a 4-vector flow W 4 analogously to the singularity of the 3-velocity w in 2-d reconnection. The final result are two flux tubes which have non-vanishing twist (not shown in the figure) and non-vanishing total helicity although the initial flux tube had zero total helicity. 3-d reconnection in general produces helicity. How much helicity is created in 3-d reconnection? Page 23 of 36

24 General considerations For a non-ideal evolution E + v B = N the change of the total helicity in a magnetically closed volume is given by d A B d 3 x = 2 N B d 3 x, dt V if N vanishes on the boundary. The total helicity is strictly conserved for ideal MHD or more general for N B = 0, e.g. in case of 2-d reconnection. The total helicity is approximately conserved on the time scale of energy dissipation for a resistive plasma [Berger 1984]. H H t with τ d = L 2 /η and L = A τ d B = A B B 2 If t τ d then H/H 1. V Page 24 of 36

25 7.2. Production of helicity in reconnection Astrophysical plasmas differ from many technical plasmas in the size of the regions V diss, where the dissipation dominates the evolution compared to the volume V of magnetic flux connected to V diss. Especially d L for astrophysical plasnas, where d is the diameter of V diss and L of V. τ hel = V A B d3 x V E Bd3 x B V L with L = A E V rec B, τ diss = V B2 /(8π)d 3 x V E Jd3 x B V d with d = B E V diss J Vdiss τ hel τ diss L d 1 In plasmas with high magnetic Reynolds numbers the helicity production in a single reconnection event is small compared to the potential helicity of the field(e.g. a corresponding constant-α force-free field ([Hornig 1999]). Page 25 of 36

26 How is the helicity created which we observe on the solar surface? A principal method: An initially untwisted flux tube with vanishing total helicity is twisted and......reconnected into two flux tubes with negative and positive total helicity. Page 26 of 36 Reconnection does not produce helicity but separates helicity!

27 7.3. Creation of helicity in the Sun In the interior of the sun, the equator rotates faster than the poles. Differential rotation provides a strong source of helicity injection (from [Berger 2000]). Transfer into the sun from magnetogram and solar rotation data transfer into the southern interior (predominantly positive curve)and northern interior (predominantly negative curve). The units are 1040 Mx2 /day. The differences in magnitude between the two curves go up to 5 x 1042 Mx2 /day Page 27 of 36

28 8. Astrophysical magnetic fields often have a complex topological structure. Topological invariants are important in the dynamics of fluids, e.g. the magnetic helicity is a better invariant then magnetic energy. Thus we need: A better description of the complexity of field structures, i.e. higher order topological invariants. C. Mayer, H.v. Bodecker Additional knowledge where and under which condition the topological structure of fields changes, i.e. about reconnection and related processes. S.V. Titov, E. Tassi, J. Kleimann Page 28 of 36

29 References [Arnold 1986] Arnold, V.I., 1986,The asymptotic Hopf invariant and its application, Sel. Math. Sov., 5, 327 [Berger 1996] Berger, M.A.,1996 Inverse cascades in a periodic domain, Astrophysical Letters & Communications, 34, 225 (1996). [Berger 1984] Berger, M.A., 1984, Rigorous new limits on magnetic helicity dissipation in the solar corona, Geophys. Astrophys. Fluid Dynamics, 30, 79 [Berger 2000] Berger, M.A., Ruzmaikin, A.,Rate of helicity production by solar rotation, J. Geophysical Research 105, (2000) [Hornig & Rastätter, 1997] Hornig, G., and L. Rastätter, The role of in the Reconnection Process, Adv. Space Res. 19, 1789, 1997a. [Hornig 2000] Hornig, G., The Geometry of Reconnection, in An to the Geometry and Topology of Fluid Flows, Kluwer gh/publiste/gtf.ps.gz. [Hornig 1999] Hornig, G., In: Brown, M.R., Canfield, R.C., Pevtsov, A.A.(eds.), in Space and Laboratory Plasmas, Geophysical Monographs, AGU, 157 [Freedman 1988] Freedman, M.H., 1988, Journal of Fluid Mechanics, 194, 549 [Freedman & He 1991a] Freedman, M.H., He, Z-X., 1991, Topology, 30, 283 [Freedman & He 1991b] Freedman, M.H., He, Z-X., 1991, Annals of Mathematics, 134, 189 Page 29 of 36

30 [Berger 1993] Berger, M.A., 1993, Physical Review Letters, 70, 705 [Moffatt 1969] Moffatt H K 1969,Journal of Fluid Mechanics Page 30 of 36

31 Covariant helicity The helicity density together with the helicity current form the helicity 4-vector h α = ɛ αβγδ A α F β γ = (A B, ΦB + E A) and the balance equation reads ɛ αβγδ δ A α F βγ = F αδ F βγ or in differential forms: d(a F ) = F F Page 31 of 36

32 9.2. in a periodic domain A box IR 3 with periodic boundary conditions on all sides is topologically a 3-torus and therefore different from any domain in IR 3. Especially it allows for magnetic fields which do not have a vector potential (see [Berger 1996]). There are two ways to ensure the existence of a vector potential in such a domain: Use a periodic A in your numerical integration and derive B from A or... Make sure that the magnetic flux through every side of the box vanishes. An example of what can happen in a periodic domain: Consider a magnetic field with a constant component in z-direction and an evolution of x-y components as shown below (from [Berger 1996]). Page 32 of 36 The helicity of the field changes its sign from a) to f)! Note that the reconnection process involved can be chosen such that it produces an arbitrary small change of helicity.

33 9.3. Equivalence of boundary conditions It is obvious that A n V = 0. can always be achieved by gauge transformations if B = 0 outside of an arbitrary integration volume V. Under the less restrictive assumption B n V = 0, where V is a simply connected volume, this requirement can still be fulfilled, as we will see now: Let us assume that A has a non-vanishing component A tangential to the surface V. We can express A as a one-form α defined only on V. Then the assumption B n V = ( A) n = 0 written in differential forms reads dα = 0 on V. From V being simply connected it follows that V has the same homotopy type as the two sphere S 2, but since the cohomology vector space H 1 (S 2 ; IR) = 0, all closed one-forms are exact. Therefore there exists a scalar function ψ on V such that α = dψ. This in turn implies that a gauge exists such that A V = 0 and thus A n V = 0 Page 33 of 36

34 9.4. Time dependent gauge of the vector potential We use a gauge defined by such that { Φ Φ = Φ t Ψ A à = A + Ψ, dψ dt = Φ W A Φ v à = 0. Thus A is transported in the plasma flow v as a 1-form: t A + (v A) v A = 0 in terms of a Lie-derivative for the 1-form A t A + L v A = 0 Page 34 of 36

35 9.5. Covariant description If the effect of the non-ideal term N is limited to isolated regions, embedded in an ideal environment ( N = 0) then the non-ideal flow v can be mapped to an ideal 4-velocity W (4) = (W 0, W 1, W 2, W 3 )= (W 0, W) satisfying ɛ αβγδ α W ν F νβ = 0 { 0 (W 0 E + W B) + (E W) = 0 W 0 0 B (W B) W 0 E = 0 L W ωf 2 = 0 F µν dx µ dx ν = const. (2) C where F αβ is the electromagnetic field tensor ([Hornig 2000]. Special solutions: W 0 E + W B = 0 W/W 0 = x s / ct s = w The 4-velocity W 4 vanishes at the reconnection line (w is singular in this case). The covariant transport of the electromagnetic field tensor generally does not imply the conservation of magnetic flux. Exception: W 0 is constant or E B = 0. Page 35 of 36

36 9.6. Interpretation For systems of (untwisted) flux tubes the total magnetic helicity can be expressed as a sum over the mutual linking of flux tubes [Moffatt 1969]: H(B) = 2 i<j lk(t i, T j )Φ i Φ j, where lk(t i, T j ) is the linking number of the tube T i and T j with magnetic fluxes Φ i and Φ j. This interpretation was generalized by [Arnold 1986] for the generic case where field lines are not closed using asymptotic linking numbers. Note: Twist is a linkage of sub-flux tubes: c) Page 36 of 36