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?