1. 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

2. What happens after an intensive time if new perspectives remain unclear (E.Much, Tag danach)

3. Bufferfly diagramm for current helicity as observed at Huairou

4. 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

5. 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.

6. 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

8. 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

9. 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.

10. Every two lines are non-linked, however 3 are linked.

11. Let us deal for the time being with polynomial invariants only. A problem: density of the invariant + of course, all other problems

12. It remains unclear how to define the density

13. 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.

14. a c Mutual helicity

15. Helicity invariants Pixel size11/21/3 # tubes1639658214716 H 1 (10 9 )2.3782.3792.323 H 2 (10 -3 )4.4764.8034.737

16. 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?