As dened by the equation (1.22) the magnetic helicity in a given volume, is the volume integral of the dot product of magnetic vector potential (A) and the magnetic eld. As the curl of a vector measures its rotation around a point, this relation gives how much A rotates around itself times its own modulus i.e. like a helix. The name helicity is thus appropriate as it gauges the relative curling or braiding of the lines of A and b (i.e. to what degree they resemble helixes). This could be termed as the curliness of the eld [24]. Several types of magnetic elds including twisted, kinked, knotted or linked magnetic ux tubes, sheared layers of magnetic ux and force-free elds, all possess magnetic helicity. As magnetic helicity quanties various aspects of the magnetic eld structure, it allows the comparison of models of elds in dierent geometries, avoiding the use of parameters specic to the model [25].
Thus it is a topological property of the magnetic eld and can be measured as follows:
consider three ux tubesT1, T2 andT3 with uxesφ1,φ2 andφ3 respectively, interlinked as shown in the gure 1.1. Here ux φ of a magnetic eld b is dened as the surface integralR
Sb.dS across a surfaceS(t) bounded by a closed curvel(t), which is moving with the plasma. Sweeping the boundary curve l along the eld denes a ux tube [1]. From this conguration, magnetic helicity is determined using the Gauss linking number L(Ti,Tj) or simply Lij3 [25] between any two ux tubes Ti and Tj, of any N
3Gauss linking number determines the twist between two ux tubes. and in equation (1.27) the term 2Lijφiφj determines the mutual helicity between the two ux tubes. When i = j, the term determines the self helicity withLii representing an average twistTi within a ux tube.
1.2 Magnetic Helicity 11
Figure 1.1: Schematic of linking of the ux tubes. Shown are three ux tubesT1, T2andT3interlinked together, with their respective uxesφi, i=1-3. Adapted from [3].
ux tubes, from [3] as math-ematical operations shown below for two ux tubes, as in [25, 3]. Letσ parameterize the curve 1 and τ parameterize the curve 2, with points x(σ) and y(τ) on the each of the curves respectively. Letr=y−x. From the denition of Gauss linking number:
L12 =− 1 Combining this equation and equation (1.27),
HM =−1 4π
Z Z
b(x). r
r3b(y)d3x d3y.
Simplifying the calculations using Coulomb gauge for the vector potential (∇ ·A=0):
A(x) = − 1 which is same as equation (1.22).
1.2.1 Ideal Invariance of Magnetic Helicity
From the denitions in equations (1.22) and (1.26), which have similar type of terms, for helicities, it might be tempting to declare magnetic helicity as an ideal invariant in 3D-MHD, like kinetic helicity in 3D-HD. The vector potential, is a gauge dependent quantity, therefore the magnetic helicity is also gauge dependent. In order that magnetic helicity be an ideal invariant, it needs to be gauge invariant.
In order to prove the gauge invariance of magnetic helicity, the notation used so far is slightly modied as follows, closely in the lines of [24]. Let a divergence-free magnetic eld B(r) be given in a region D, which may be either bounded or not. Its magnetic helicity when B=∇ ×A, is dened as
h(B,D) = Z
D
A·Bd3r (1.28)
HereAis the magnetic vector potential. It is necessary to assign boundary conditions to B. It will be assumed that the magnetic eld B is parallel to the surface ∂D that bounds D. If now n is a unit vector normal to ∂D, then it is seen that B·n =0 in
∂D. As the conguration consists of only nite energy elds, B = 0 in any part of
∂D→ ∞. For the same reason a gauge transformation is considered as follows:
A0 =A+∇f. (1.29)
With the boundary conditions stated above and ∇ ·B=0, it is found that h0−h =
It is necessary that f be single valued in D for the above relation to be true. Hence magnetic helicity is a property of the transverse or solenoidal part of A. It does not depend on the longitudinal part of the magnetic vector potential (A||) ( as A is ex-pressed as the gradient of a potential which is single valued in D). If it were not so, then Gauss theorem could not be applied toR
D∇f.bd3r, the change in helicity under gauge transformation for equation (1.29) [24].
For gauge invariance the normal component ofB i.e. Bn must vanish at the boundary surface since function f is arbitrary. Only in special cases, for example if periodic boundary conditions are used, then a nite Bn is possible. Many magnetic congura-tions of interest in astrophysics are either open with eld lines extending upto innity
1.2 Magnetic Helicity 13 or bounded by surfaces crossed by eld lines. In such cases magnetic helicity is no longer gauge invariant. In order to overcome this situation an alternative formulation is needed [1]. Hence following [1] let
halt = Z
V
dV(A+A0)·(B−B0) where B0 =∇ ×A0 is a reference eld.
(1.31) The reference eld is chosen suitably. In an open system this reference eld may be a static eld with the same asymptotic properties as B. In a bounded system the normal components of both the elds should be equal. To show that halt is indeed gauge invariant even under separate gauge transformations of Aand A0, consider the conservation law for the magnetic helicity. The gauge is chosen such that the scalar potential vanishes, E =−∂tA/c, where c is the velocity of light. Applying Faraday's law
is inserted into equation (1.33), and the boundary conditionBn=0 is applied, then the second term in the equation (1.33) becomes:
− I
(A·B)v·dS . (1.35)
Hereσ is the electrical conductivity of the medium, and it is related to the dimension less magnetic diusivityηˆ(=Rm−1) throughη=c2/4πσwhich isη/L0V0 (see equation (1.13)). From the above equations the time variation of h is obtained as in equation (1.36) by using the equation (1.37) which represents the change inh due to change in dVof the volume.
I it can be shown that halt is also conserved.
First term on the r.h.s. of equation (1.33) and the r.h.s. of equation (1.36) are the two terms that depict the variation of helicity, with respect to time. These two terms represent the nonlinear helicity transmission and dissipation respectively, constituting the helicity ux and hence will be used in chapter 4 in their Fourier transformed forms.
Importance of the Invariance Property
From the above discussion, it can be inferred that the magnetic helicity is con-served in ideal MHD and is approximately concon-served during magnetic reconnection (A process in which there is a change of magnetic connectivity of plasma elements due to the presence of a localized diusion region where ideal MHD breaks down. [26]). In a conned volume, widespread reconnection may reduce the magnetic energy of a eld while approximately conserving its magnetic helicity [4]. As a result, the eld relaxes to a minimum energy state, often called the Taylor state, where the current is parallel to the force free eld [4]. Such relaxation processes are important to both fusion (es-pecially in reversed eld pinch devices) and astrophysical plasmas [25]. The derivative of magnetic helicity obtained in the equation (1.33) has two terms, a dissipative and a transport term. The dissipative term represents the eect of twisting motions on the boundary while the second transport term represents the bulk transport of helical eld across the boundary. From these two terms some astronomical observations e.g.
hemispheric specic sign of helicity4, production of solar coronal mass ejections5; could be interpreted [27, 28]. The constraint of magnetic helicity preservation implies that a dynamo (the mechanism whereby electric currents within an celestial body generate a magnetic eld) is more easily produced if the electric potential varies in the surface of the dynamo [29, 30].
The invariance constraint also infers that with external forcing or with any kind of agitations to the system, only scale changes could be achieved in the system but the magnetic helicity cannot be destroyed. Thus in order that this constraint be fullled, the magnetic eld topology in a system must change signicantly, while the total mag-netic helicity of the system remains invariant or approximately invariant. This feature
4In the Sun, observations of magnetic helicity indicate that it has a positive sign in the southern hemisphere and a negative sign in the northern hemisphere
5Huge violent ejections of plasma coming out of the Sun's outer surface i.e. corona.
1.3 Inverse Cascade of Magnetic Helicity 15