Some new formulae for magnetic and current helicities

Jean-Jacques AlyDSM/IRFU/SAp, CE Saclay, France

1. Statement of the problem• Consider a regular magnetic field B occupying the simply-

connected domain D. Denote as S the boundary of D and as n the inner normal to S. Assume that S is connected and all the magnetic lines of B cut S.

• Let Bp be the potential field satisfying Bpn=Bn on S, and introduce the arbitrary vector potentials A and Ap such that B=∇xA and Bp=∇xAp. If D is unbounded, assume that B, Bp, A, and Ap, decrease sufficiently fast at infinity.

• Then the relative magnetic helicity of B is defined by

(Berger & Field 1984, Finn & Antonsen 1985).

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H = (A + Aπ∫ ).(B − Bπ )dv

• Convenient gauges: B = ∇xC and B = ∇xCp , with

Cp uniquely defined, C defined up to +∇f, with f=k on S.

• With these gauges:

• Evolution of H in ideal MHD: if plasma moves at velocity v on S, then

(a subscript « s » indicates a component parallel to S).• If u = 0, H keeps a constant value.

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∇.Cπ = 0 in D, Cπn = 0 and n × (C − Cπ ) = 0 on S.

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H = C.BdvD

∫ .

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dH

dt= −2 u.Cπ Bndv

S

∫ , where u = vs − (vn /Bn )Bs.

• Then two fields which have the same line topology and the same Bn on S, have also the same helicity. Indeed they can be transformed into each other by ideal motions keeping fixed the positions of the footpoints on S and thus satisfying u=0.

• This implies that H depends only on the topology of the lines of B in D and on Bn on S.

• This leads to the following natural question:

Is it possible to write an expression of H in which this dependence on line topology and on Bn appears explicitly?

2. Ingredients• S = S+ S⋃ - S⋃ 0, with Bn >0/<0/=0 on S+/-/0. S0 is a curve (PIL).

• Our starting point will be the following formula for relative magnetic helicity (Berger 1988):

The quantity h is called line helicity by Berger and topological flux function by Yeates & Hornig (2012).

• Note that h is invariant by the gauge transforms allowed for C (C C⇾ +∇f, with f=k on S). On the other hand, h keeps the same value if B suffers an ideal MHD transform with u=0 on S (Berger 1988).

• Our other main ingredient will be the magnetic mapping of B.

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H = C.BdvD

∫ = C.dlL(r )

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟

S +

∫ Bnds = hBn

S +

∫ ds, where h(r) = C.dlL(r )

∫ .

• Magnetic mapping:– The magnetic lines L of B in D define the magnetic mapping

M : 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 its footpoint on S-.

– In most cases, M is discontinuous accross some curves Gj S⊂ +: two infinitely close points r1 and r2 located on each side of Gj have images M(r1) and M(r2) separated by a finite distance. The magnetic lines connected to Gj form a singular surface in D, a so-called separatrix, which either contains a neutral point of B (where B=0) or is tangent to S along a so-called bald patch S⊂ 0.

– The domain S+/(G1+G2+…) decomposes into p components S+k

inside which M is continuous. We note S-k the domain

M(S+k) S⊂ -.

3. Helicity of a simple topology field• Assume that B has simple topology (M is continuous on

S+, there are no separatrices). • Consider in D the tubular

magnetic surface S cutting S+

along the arbitrary oriented closed curve C+ and S-

along C-=M(C+). Then (no flux through S + Stokes)

where M* = linear mapping tangent to M, Ĉ(r)=C[M(r)], and we have used that Cs = Cπs on S. €

C.dlC +

∫ = C.dlC −

∫ ⇒ dlC +

∫ .(Cπs − M* . ˆ C πs) = 0 ⇒ ∃χ s.t. -∇sχ = Cπs − M*. ˆ C πs,

S+ C+

C-

S

S-S0

• Next consider the magnetic surface S bounded by the arbitrary oriented curve C+ S⊂ +

connecting the points r1 and r, its image C- = M(C+) ⊂S-

, and the two magnetic lines L(r1) and L(r). By the same token

• We choose (c r1)=0 and r1 on S0, whence h(r1)=0.

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C.dlC +

∫ + C.dlL( r)

∫ − C.dlC -

∫ − C.dlL( rk )

∫ = 0 ⇒ h(r) − χ (r) = h(r1) − χ (r1).

S+

C+

C-

S

S-S0r1 r

L(r1)

L(r)

• We can thus write

and

The second expression is obtained by an integration by parts. To get the third one, we introduce arbitrary coordinates (x1,x2) on S+ and (X1,X2) on S-, and express the magnetic mapping as M: (x1,x2) (X↦ 1(x1,x2),X2(x1,x2)) (ekj is the 2D alternating tensor).

• Then we have reached our goal for a simple topology field: H has been expressed in terms of Cp, which depends only on Bn, and of the topology of the lines, which is determined by the magnetic mapping.

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H = χS +

∫ Bnds = (Cπs × M*. ˆ C πs)S +

∫ .nds = ε kj∫ Cπkˆ C πi∂ j X

idx1dx 2.

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h(r) = χ (r) = dl.(CπsC( r1 ,r )

∫ − M*. ˆ C πs).

4. Helicity of a complex topology field

• If B has a complex topology, we can do the same construction in each domain S+

k.• We thus have

(where we have imposed c(rk)=0) and

where Φk = magnetic flux through S+k . Note that the

integration by parts in the second expression has brought in a new term – an integral along the boundary of S+

k which vanished in the previous case due to c=0 on ∂S+.

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for r ∈ Sk+ : h(r) = h(rk ) + χ (r), with χ (r) = dl.(C πs

C( rk , r )∫ − M*. ˆ C πs),

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H = h(rk )Φk + χSk

+

∫ Bnds ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥k

∑ = h(rk )Φk + (Cπs × M* . ˆ C πs)Sk

+

∫ .nds + χCπs.dl∂Sk

+

∫ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥k

∑ ,

• To reach our goal, we need 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: then we can choose rk on ∂k and set h(rk)=0.

– Or this is not the case and we need to relate h(rk) to the topology and to the flux distribution on the boundary. In that case the following problem arises: is the topology uniquely determined by the magnetic mapping, or it is necessary to introduce some supplementary parameters to characterize it? There are still some points to clarify to answer these points in full generality.

• Let us just here illustrate these problems by discussing some quite simple examples.

– Consider fields constructed as follows: in each plane x=const, they coincide with the field created by two 2D dipoles, one of moment m(x) located at (y=-d,z=-h(x)), and one of moment n(x) located at (y=d,z=-h(x)). By a simple adjustement of the parameters, one may get configurations with the following structure on S:

Green: PIL Blue: bald patch Red: trace of separatrix

In both cases, all the h(rk) can be taken to vanish. But contribution of the Gk to the line integral in the second formula.

S+

S-G

S -2

S+1

S-1

S-2

S+4

S-3

S+2

S+3

S-4

G1

G2

• Consider next a quadrupolar field in the exterior of a spherical domain. One can take h(r2)=h(r3)=h(r4)=0, and h(r1)=ha+hc-hb. Here the magnetic mapping fully determines the topology and all the h(rk) can be determined.

• Consider finally a field in a cylinder which is obtained by the ideal MHD deformation of a uniform vertical field (Parker’s model). Here S0 has a finite area, and it is clear that the magnetic mapping does not determine the topology as one may give an overall twist of 2pn without changing it. It is possible however to compute h(r1), with r1 located on the boundary of S+: in the simplest case where the outer lines have a twist 2pn, h(r1)=nF.

+1+2

-2-3

+3+4

-4-1

a

b

c

S+

S0

S-

B0

B

v

v

D

5. Magnetic energy and current helicity when B is a force-free field

• Assume that B is force-free, i.e., it does satisfy in D the equation x∇ B = αB, with α = const. along any line L.

• B has energy W and current helicity Hc, with

• Because electric currents are flowing along the lines, g can be computed in the same way as h, by just substituting Bs for Cπs.

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W =1

8πB2

D

∫ dv =1

8πBdl

L( r)

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟Bn

S +

∫ ds =1

8πgBn

S +

∫ ds, where g(r) = BdlL( r)

∫ ,

Hc = B.∇ × BdvD

∫ = αB2

D

∫ dv = α BdlL( r)

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟Bn

S +

∫ ds = αgBn

S +

∫ ds.

• Then:

and

where (c/4p)Ik = total current through S+k. Of course all

the considerations on the computation of h(rk) apply to g(rk).

• The energy can be also expressed in the form

where the first equality is due to Berger (1988).

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for r ∈Sk+ : g(r) = g(rk ) +ψ (r), with ψ (r) = dl.(Bs

C( rk ,r )

∫ − M*.ˆ B s),

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W =1

8πg(rk )φk + ψBnds

Sk+

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

k

∑ , (*)

Hc = g(rk )Ik + αψBndsSk

+

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

k

∑ ,

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W =1

8παhds + (Cπs × Bs).nds

S

∫S +

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟=

1

8πh(rk )Ik + αχBnds

Sk+

∫ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+ (Cπs × Bs).nds

S

∫k

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥,

• Energy can also be expressed in terms of B on S by using the virial theorem. In the case of the Parker’s model (force-free field in a cylindrical domain of height h), the virial relation can be combined with relation (*) above to derive a rigorous upper bound on the energy of a force-free field having a given topology. For instance, in the simplest case where Bn=B0 and the outer lines are not twisted, one gets

where R=(x,y) and Rh=M-hz.

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W −Wπ ≤B0

2

3hR.∇Rh (R) − Rh (R)

2ds

S +

∫ ,

6. Conclusion

• We have obtained new expressions in which the dependence of the relative magnetic helicity on the topology of the lines and the flux distribution on the boundary appears explicitly.

• We have also derived new expressions for the magnetic energy and current helicity of a force-free field in which intervene the field on the boundary and the topology of the lines.

• Question: can these formulae be useful in solar physics?

Thanks for your attention