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D.Sokoloff,Egor Illarionov Moscow State University, Russia Alexander Smirnov, Peter Akhmetyev IZMIRAN, Helicity Thinkshop on Solar Physics October 27-31, 2013, Beijing, China After current helicity: higher topological invariants
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What happens after an intensive time if new perspectives remain unclear (E.Much, Tag danach)
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Bufferfly diagramm for current helicity as observed at Huairou
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Why helicity data are instructive? Current helicity – observable, clear topological meaning – linkage of current Mirror asymmetric Magnetic helicity – topological invariant – linkage of magnetic lines Reconnections are slow – inviscid invariant of motion Can be in principle obtained from observations
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Difficulties Magnetic tube is a not very elaborated concept. Magnetic line covers often a 3D domain. How to resolve: Arnold suggested a technique of short ways. We do not know magnetic field inside vthe Sun (and in some other domains as well). How to resolve: relative helicity. Linkages in respect to a given field.
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How to use? Magnetic field relaxation – is cxontrolled by magnetic helicty Dynamo: dynamo generated mean field is helical. One have to conserve total helecity. IMPORTANT: Magnetic helicity can not be transported along the spectrum: ab \sim aa/k >> vv for small k. Upper bound for helicity (no helicity without energy). Capacity of higher Levels is insufficient
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Magnetic helicity densityis gauge non-invariant How to resolve: A natural gauge follows from the local homogeneity and isotropy and axial symmetry One can calculate magnetic helicity from the current one
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Magnetic helicity: algebraic sum of linkages A cancellation of linkages can happen. Many other invariants are possible – say, sum of squares of linkages. Never vanishes if the field is linked. Polynomial invariants.
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Every two lines are non-linked, however 3 are linked.
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Let us deal for the time being with polynomial invariants only. A problem: density of the invariant + of course, all other problems
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It remains unclear how to define the density
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Helicity butterfly diagrams: predicted, expected and observed. Very noisy as expected. Two observed diagrams are very similar! - effective suppression of the noise. Helicity patterns are similar in general to the sunspots pattern! Theory predicts a wrong time lag Unexpected areas of the «wrong» helicity sign.
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a c Mutual helicity
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Helicity invariants Pixel size11/21/3 # tubes1639658214716 H 1 (10 9 )2.3782.3792.323 H 2 (10 -3 )4.4764.8034.737
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Conclusions: 1. Higher (polynomial) invariants are in principle measurable from available data. 2. This very invariant is mirror- symmetric (not very interesting for dynamo). To be clarified: 1.Can these invariants be transported along the spectrum? 2. What about other invariants?
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