Some new formulae for magnetic and current helicities

1. Some new formulae for magnetic and currenthelicities Jean-Jacques Aly DSM/IRFU/SAp, CE Saclay, France

2. 1. Statement of the problem • Consider a regularmagneticfieldBoccupying the simply-connecteddomain D. Denote as S the boundary of D and as n the inner normal to S. Assume that S isconnected and all the magneticlines of Bcut S. • Let Bpbe the potentialfieldsatisfyingBpn=Bn on S, and introduce the arbitraryvectorpotentialsA and ApsuchthatB=∇xA and Bp=∇xAp. If D isunbounded, assume that B, Bp, A, and Ap, decreasesufficientlyfastatinfinity. • Then the relative magnetichelicityof Bisdefined by (Berger & Field 1984, Finn & Antonsen 1985).

3. Convenient gauges: B = ∇xC and B = ∇xCp , with Cpuniquelydefined, Cdefined up to +∇f, with f=k on S. • Withthese gauges: • Evolution of H in ideal MHD: if plasma moves atvelocityv on S, then (a subscript « s » indicates a component parallel to S). • If u = 0, H keeps a constant value.

4. Thentwofieldswhich have the same line topology and the sameBn on S, have also the samehelicity. Indeedtheycanbetransformedintoeachother by ideal motions keepingfixed the positions of the footpoints on S and thussatisfyingu=0. • This impliesthat H dependsonly on the topology of the lines of B in D and on Bn on S. • This leads to the followingnatural question: Is it possible to write an expression of H in whichthisdependence on line topology and on Bnappearsexplicitly?

5. 2. Ingredients • S = S+ ⋃ S- ⋃ S0, withBn>0/<0/=0 on S+/-/0. S0 is a curve (PIL). • Our starting point willbe the following formula for relative magnetichelicity (Berger 1988): The quantity h iscalledline helicityby Berger and topological flux functionby Yeates & Hornig (2012). • Note that h is invariant by the gauge transformsallowed for C (C ⇾ C+∇f, with f=k on S). On the other hand, h keeps the same value if Bsuffers an ideal MHD transformwithu=0 on S (Berger 1988). • Our other main ingredientwillbe the magneticmappingof B.

6. Magneticmapping: • The magneticlinesL of B in D define the magneticmappingM : S+⟼ S- . The latter associates to the position r of the footpoint of the line L=L(r) on S+ the position M(r) of itsfootpoint on S-. • In most cases, MisdiscontinuousaccrosssomecurvesGj⊂S+: twoinfinitely close points r1 and r2located on eachside of Gj have images M(r1) and M(r2) separated by a finite distance. The magneticlinesconnected to Gjform a singular surface in D, a so-calledseparatrix,whicheithercontains a neutral point of B (whereB=0) or is tangent to S along a so-calledbald patch ⊂S0. • The domain S+/(G1+G2+…) decomposesinto p components S+kinsidewhich M iscontinuous. We note S-kthe domain M(S+k)⊂S-.

7. 3.Helicity of a simple topologyfield • Assume thatB has simple topology (Miscontinuous on S+, there are no separatrices). • Consider in D the tubularmagnetic surface Scutting S+along the arbitraryorientedclosedcurve C+ and S-along C-=M(C+). Then (no flux throughS + Stokes) whereM* = linearmapping tangent to M, Ĉ(r)=C[M(r)], and we have usedthatCs = Cπs on S. S C- S0 S- S+ C+

8. L(r1) L(r) • Nextconsider the magnetic surface Sbounded by the arbitraryorientedcurveC+⊂S+connecting thepoints r1 and r, its image C- = M(C+) ⊂ S-, and the twomagneticlinesL(r1) and L(r). By the sametoken • Wechoosec(r1)=0 and r1 on S0, whence h(r1)=0. r r1 S C- S0 S- S+ C+

9. Wecanthuswrite and The second expression isobtained by an integration by parts. To get the third one, weintroducearbitrarycoordinates (x1,x2) on S+ and (X1,X2) on S-, and express the magneticmapping as M: (x1,x2) ↦ (X1(x1,x2),X2(x1,x2)) (ekjis the 2D alternatingtensor). • Thenwe have reachedour goal for a simple topologyfield: H has been expressed in terms of Cp, whichdependsonly on Bn, and of the topology of the lines, whichisdetermined by the magneticmapping.

10. 4.Helicity of a complextopologyfield • If B has a complextopology, wecan do the same construction in eachdomainS+k. • Wethus have (wherewe have imposedc(rk)=0) and whereΦk = magnetic flux throughS+k . Note that the integration by parts in the second expression has brought in a new term – an integralalong the boundary of S+kwhichvanished in the previous case due to c=0 on ∂S+.

11. To reachour goal, weneed to compute the numbers h(rk). Two cases are possible: • Either ∂S+k and ∂S-k have a common part ∂k (included in the PIL S0) over which the lines are bridging: thenwecanchooserk on ∂k and set h(rk)=0. • Or thisis not the case and weneed to relate h(rk) to the topology and to the flux distribution on the boundary. In that case the followingproblem arises: is the topologyuniquelydetermined by the magneticmapping, or itisnecessary to introducesomesupplementaryparameters to characterizeit? There are stillsome points to clarify to answerthese points in full generality. • Let us justhereillustratetheseproblems by discussingsomequite simple examples.

12. G S- G1 • Considerfieldsconstructed as follows: in each plane x=const, theycoincidewith the fieldcreated by two 2D dipoles, one of moment m(x) locatedat (y=-d,z=-h(x)), and one of moment n(x) locatedat (y=d,z=-h(x)). By a simple adjustement of the parameters, one mayget configurations with the following structure on S: Green: PIL Blue: bald patch Red: trace of separatrix In both cases, all the h(rk) canbetaken to vanish. But contribution of the Gk to the line integral in the second formula. S-2 S+2 S-3 G2 S-2 S+3 S+ S+1 S+4 S-4 S-1

13. a +1 +2 -2 b • Considernext a quadrupolarfield in the exterior of a sphericaldomain. One cantake h(r2)=h(r3)=h(r4)=0, and h(r1)=ha+hc-hb. Here the magneticmappingfullydetermines the topology and all the h(rk) canbedetermined. • Considerfinally a field in a cylinderwhichisobtained by the ideal MHD deformation of a uniform vertical field (Parker’s model). Here S0 has a finite area, and itisclearthat the magneticmappingdoes not determine the topology as one maygive an overall twist of 2pn withoutchangingit. It is possible however to compute h(r1), withr1located on the boundary of S+: in the simplest case where the outerlines have a twist 2pn, h(r1)=nF. -3 +3 v S- c +4 -4 B -1 S0 D B0 v S+

14. 5. Magneticenergy and currenthelicitywhenBis a force-freefield • Assume thatBisforce-free, i.e., itdoessatisfy in D the equation∇xB = αB, withα = const. alongany line L. • B has energy W and currenthelicityHc, with • Becauseelectriccurrents are flowingalong the lines, g canbecomputed in the sameway as h, by justsubstitutingBs for Cπs.

15. Then: and where (c/4p)Ik = total currentthroughS+k. Of course all the considerations on the computation of h(rk) apply to g(rk). • The energycanbealsoexpressed in the form where the first equalityis due to Berger (1988).

16. Energycanalsobeexpressed in terms of B on S by using the virialtheorem. In the case of the Parker’s model (force-freefield in a cylindricaldomain of height h), the virial relation canbecombinedwith relation (*) above to derive a rigorousupperbound on the energy of a force-freefieldhaving a giventopology. For instance, in the simplest case whereBn=B0 and the outerlines are not twisted, one gets whereR=(x,y) and Rh=M-hz.

17. 6. Conclusion • We have obtained new expressions in which the dependence of the relative magnetichelicity on the topology of the lines and the flux distribution on the boundaryappearsexplicitly. • We have alsoderived new expressions for the magneticenergy and currenthelicity of a force-freefield in whichintervene the field on the boundary and the topology of the lines. • Question: cantheseformulaebeuseful in solarphysics?

18. Thanks for your attention