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D.Sokoloff,Egor IllarionovMoscow State University,
RussiaAlexander Smirnov, Peter
AkhmetyevIZMIRAN,
Helicity Thinkshop on Solar PhysicsOctober 27-31, 2013, Beijing, China
After current helicity: higher topological invariants
What happens after an intensive time if new perspectives remain unclear (E.Much, Tag danach)
Bufferfly diagramm for current helicity as observed at Huairou
Why helicity data are instructive?
Current helicity – observable, clear topological meaning – linkage of current
Mirror asymmetric Magnetic helicity – topological invariant –
linkage of magnetic linesReconnections are slow – inviscid invariantof motionCan be in principle obtained from
observations
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.
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 >> vvfor small k. Upper bound for helicity (no helicity
without energy). Capacity of higherLevels is insufficient
Dy n am o ac t ion : s t re t c h , t w is t , fo ld
Ze ldovichKr a kow, 1972
F r oze n -in m a g n e t ic fie ld
Magnetic helicity densityis gauge non-invariant
How to resolve: A natural gaugefollows from the local homogeneity and isotropy and axial symmetry
One can calculate magnetic helicity from the current one
Magnetic helicity: algebraic sum of linkagesA 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.
Every two lines are non-linked, however 3 are linked.
Let us deal for the time being with polynomial invariants only.A problem: density of the invariant+ of course, all other problems
It remains unclear how to define the density
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
.
)(21
ccacL
a
c
Mutual helicity
Helicity invariants
Pixel size 1 1/2 1/3
# tubes 1639 6582 14716
H1 (109) 2.378 2.379 2.323
H2 (10-3) 4.476 4.803 4.737
c,a
caac1 LH ca
caacLH,
22
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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