Magnetic helicity of solar active regions

Transcription

1 Magnetic helicity of solar active regions J. K. Thalmann Solar Group Seminar 12 Jul, 2011

2 Helicity of interlinked curves Number of enlacements of two interlinked curves l and l (Gauss, C. F., 1833, Werke, 5, 605.) L ll = 1 4π l l ( dl dl ) r r r r 3 (1) L... Gauss linking number r, r... points on the curves l and l dl, dl... line segment of l and l - remains the same if the two curves are interchanged - is left unchanged if l or l are moved without intersecting each other If there are N curves with flux φ i, i = 1,..., N in a closed volume: H m,tot = = N N L ij φ i φ j i=1 j=1 L ij φ i φ j + L ii φ 2 i (2) i j i=j N(N 1) terms where i j : mutual helicity between curves i and j + N terms where i = j: self helicity of curves i

3 Examples L ii = 0 L ii = 0 L ii = 5 L jj = 0 L jj = 0 L ij = 1 L ij = 3 L ij = 0 H m,tot = 2 L ij φ i φ j H m,tot = 2 L ij φ i φ j H m,tot = L ii φ 2 i = 2 φ i φ j = 6 φ i φ j = 5 φ 2 i sign depends on relative orientation of flux

4 Helicity in a volume Real cases: - infinite number of field lines (concentrated in magnetic flux tubes) - complex combination of twist and linking (contributing different amounts to H m,tot ) Approximation of H m,tot in a volume (Moffat, H. K., 1969, J. Fluid Mech., 35, ) : - generalized formulation of the linkage of pairs of flux tubes and the twist of individual flux tubes L ll = 1 4π l l ( dl dl r r N ) r r 3, H m,tot = L ll φφ, φ = l,l S B ds =1 The helicity of a magnetic field B = A, in a volume V, is given by if n B = 0 on V. H m,tot = dv A B (3) V A... magnetic vector potential, A B... magnetic helicity density

5 Linkage ambiguity What is the meaning of magnetic closure (n B = 0 on V)? H m,tot = 0 H V = Φ 2 H V = H V B mirror symmetric in V H m,tot = 2 Φ 2 H V = Φ 2 H V = H V B identical in V If no magnetic closure (n B 0 on V): - magnetic field lines have endpoints on V, linking numbers are undefined - ambiguity in counting linkage of flux tubes gauge ambiguity Figures taken from Hornig, G., 2006, arxiv:astro-ph/

6 Linkage ambiguity What is the meaning of magnetic closure (n B = 0 on V)? H m,tot = 0 H V = Φ 2 H V = H V B mirror symmetric in V H m,tot = 2 Φ 2 H V = Φ 2 H V = H V B identical in V H m,tot =? B unknown in V If no magnetic closure (n B 0 on V): - magnetic flux tubes have endpoints on V linking numbers are undefined - ambiguity in counting linkage of flux tubes gauge ambiguity Figures taken from Hornig, G., 2006, arxiv:astro-ph/

7 Relative Magnetic Helicity Berger, M., A. & Field, G. B., 1984, J. Fluid Mech., 147, ; Finn, J., M. & Antonsen, T., M., 1985, Comm. Plasma Phys. Control. Fusion, 9, current-carrying field B reference field Bp solar atmosphere solar interior Helicity of a field B = A with respect to a reference field B p = A p in a volume V: H m,rel = dv (A + A p ) ( B B p ) = dv ( A B A p B p ) (4) V V if n B = n B p and n A = n A p on V. Figure adapted from Démoulin, P., 2007, Adv. in Space Res., 39,

8 Practical application NOAA AR Jan, 2004 Location on solar disk: 18-Jan-2004: S14 E15 19-Jan-2004: S14 E02 20-Jan-2004: S14 W10 21-Jan-2004: S14 W22 No need to take curvature into account Data: Solar Flare Telescope - Vector-magnetograph measures ZE on the Fe i nm spectral line

9 Practical application NOAA AR Jan, 2004 Location on solar disk: 18-Jan-2004: 19-Jan-2004: 20-Jan-2004: 21-Jan-2004: S14 E15 S14 E02 S14 W10 S14 W22 Data: Solar Flare Telescope - Vector-magnetograph 18-Jan Jan-2004 B [mt] 20-Jan-2004 B [mt] Jan-2004 B [mt] B [mt] mt x [arcsec] mt x [arcsec] mt x [arcsec] mt x [arcsec] y [arcsec] y [arcsec] 0 y [arcsec] N y [arcsec]

10 Practical application NOAA AR Jan, 2004 Data: Solar Flare Telescope - Vector-magnetograph Method: - force-free extrapolation (potential & NLFF) applicable Mm above photosphere - calculate 3D field lines - calculate magnetic vector potentials - calculate relative helicity content GOES nm SXR flux Time instances of SFT/VM data 10-4 Flux [W m -2 ] M C B Jan 19-Jan 20-Jan 21-Jan Start Time (18-Jan-04 00:00:00)

11 NOAA AR Jan, D magnetic field structure: 20-Jan-2004:

12 NOAA AR Jan, 2004 Magnetic Flux [ Wb ] F m = S ds B z(x, y, z = 0) F m F m + F m Jan Jan Jan Jan-04 Magnetic Energy [ J ] E m = 1 2µ dv B2 0 V E tot E pot E free 18-Jan Jan Jan Jan-04 Relative Helicity [ Mx 2 ] F + m F m,tot, F m F m,tot H m,rel = V dv ( A + A p ) ( B B p ) Jan Jan Jan Jan E free 0.3 E tot, E 20 free 0.24 H m,rel > 0, H m,rel E m,tot [10 27 J] H rel [10 44 Mx 2 ] F m [10 15 Wb] Jan Jan Jan Jan-04

13 Helicity proxies Hemispheric sign trend Predominantly negative/positive helicity patterns in northern/southern hemisphere 19 Jan, 2004: NOAA AR located at 14 S - differential rotation directly twists flux tube foot points - right-handed twist of field lines about its axis within flux tube - implies positive helicity Figure taken from Bao, S. D., et al., 2002, Astrophys. J., 573,

14 Helicity proxies Magnetic tongues: - elongations of magnetic polarities - observable during emergence - flux distribution interpreted as due to emergence of twisted flux tube - only two possible configurations - if sign of polarities is reversed, sign of helicity does not Figures taken from Luoni, M. L., et al., 2011, Sol. Phys., 270,

15 Helicity proxies Magnetic tongues: 18-Jan-2004

16 Summary tools for extrapolating the force-free magnetic field in the outer solar atmosphere well established methods to calculate the vector potentials implemented and tested knowledge of the vector potential allows us to calculate the relative magnetic helicity estimated relative helicity has to be compared to helicity proxies like magnetic tongues, magnetic twist,...

17 From the Gauss linking number towards magnetic helicity a b c = b c a L ll = 1 4π l l dl dl r r r r = 1 ( ) r r dl 4π l l r r 3 dl (5) H = L ll φφ, φ = S B ds dφ dl = B ds dl H m,tot = 1 4π l dφ dl l ( r r r r 3 dφ dl A(r ) = µ 0 4π V dv J(r ) r r = 4π 1 V dv B(r ) r r (fa) = f a f a ( 1 r r ) = r r r r 3 ) = 1 ( ) r r dv dv 4π V V r r 3 B(r ) B(r) (6) A(r ) = 1 dv 1 4π V r r B(r ) = 1 V r r 4π dv r r 3 B(r ) (7) H m,tot = dv A B (8) V

18 Computation of the magnetic vector potential Berger, M., A. & Field, G. B., 1984, J. Fluid Mech.; Thalmann et al. 2011, Sol. Phys., submitted: A p = 0 A = 0 B = B + B p A = A + A p with n A = n A p A p = 0 with n ( A p ) = n B p A = µ 0 J with n A = 0 DeVore, R., C., 2000, Astrophys. J.; Valori et al., 2011, Sol. Phys., submitted: z A p = 0 A p = c z z 0 dz B p(x, y, z ) A = A p (x, y, 0) z z 0 dz B(x, y, z ) z A = 0