Two interpretations of a vector potential
Recently, Berger & Hornig have published a nice paper on the poloidal-toroidal (P-T) decomposition of magnetic fields, proposing that it can be used to make a simple and elegant definition of magnetic helicity. In this blog post, I want to show that this is equivalent to the “winding gauge” helicity proposed by Prior & Yeates, for the case of a magnetic field extending between two parallel planes. (This is stated by Berger & Hornig at the bottom of page 6.) As such I will focus only on this Cartesian domain, but an important aspect of the Berger & Hornig paper is the generalization of the P-T decomposition to more general shapes of domain (see the paper for details).
Poloidal-Toroidal decomposition in Cartesian domains
The Cartesian P-T decomposition is the well-known splitting \({\bf B} = {\bf B}_P + {\bf B}_T\) where \[ {\bf B}_T = \mathcal{L}T, \qquad {\bf B}_P = \nabla\times\mathcal{L}P, \] and \(\mathcal{L}\) is the operator \(\mathcal{L}f = \nabla\times{f\hat{\bf z}}.\) The toroidal component \({\bf B}_T\) has no z-component, while the poloidal component \({\bf B}_P\) has no vertical current (i.e., \(\hat{\bf z}\cdot\nabla\times{\bf B}_P=0\)). In fact, one can think of \({\bf B}_P\) as being “generated” by \(B_z\) and \({\bf B}_T\) as being generated by \(j_z\). To see this, note that the above equations imply that T and P satisfy the two-dimensional Poisson equations \[ \Delta_\parallel T = -J_z, \qquad \Delta_\parallel P = -B_z \] in each plane of constant \(z\). It is easy to see that possible vector potentials for \({\bf B}_T\) and \({\bf B}_P\) are given by \[ {\bf A}_T = T\hat{\bf z}, \qquad {\bf A}_P = \mathcal{L}P = \nabla\times(P\hat{\bf z}). \] In terms of these vector potentials, the magnetic helicity is \[ H = \int_V{\bf A}\cdot{\bf B}\,\mathrm{d}V \\= \int_V{\bf A}_T\cdot{\bf B}_T\,\mathrm{d}V + \int_V{\bf A}_T\cdot{\bf B}_P\,\mathrm{d}V + \int_V{\bf A}_P\cdot{\bf B}_T\,\mathrm{d}V + \int_V{\bf A}_P\cdot{\bf B}_P\,\mathrm{d}V. \] Now for this \({\bf A}_T\) we always have \(\int_V{\bf A}_T\cdot{\bf B}_T\,\mathrm{d}V=0,\) i.e. a toroidal field is never linked with itself. If we consider an infinite domain sandwiched between two planes of constant z, then you can show – by substituting in the various expressions and integrating by parts – that (i) the last term also vanishes (i.e. a poloidal field has no self linking), and (ii) both cross terms are equal. Note that neither of these properties holds if the side boundary is finite rather than extending to infinity. But for an infinite boundary, the helicity in this gauge reduces to \[ H = 2\int_V\mathcal{L}P\cdot\mathcal{L}T\,\mathrm{d}V. \] This is the elegant form given by Berger & Hornig (equation 35). It shows that the helicity in this gauge may be interpreted as the linking between poloidal and toroidal components. (In fact, for this domain this gauge of \(H\) will also match the relative helicity, since the reference potential field will have \(j_z=0\) on all planes and so be purely poloidal.)
Winding gauge vector potential
Prior & Yeates proposed a different way to choose a unique helicity for the volume between two planes, by fixing the vector potential to the winding gauge defined by the two-dimensional Biot-Savart like integral \[ {\bf A}^W(x_1,x_2,z) = \frac{1}{2\pi}\int_{S_z}\frac{{\bf B}(y_1,y_2,z)\times{\bf r}}{|{\bf r}|^2}\,\mathrm{d}^2y, \] where \({\bf r} = (x_1-y_1, x_2-y_2,0)\) and \(S_z\) is the plane at height \(z\). In this gauge, \(H\) may be interpreted as the average pairwise winding between all pairs of magnetic field lines, as they thread the domain from the lower to the upper boundary.
Proof of equivalence
I will show (for the infinite domain between two horizontal planes) that \({\bf A}^W = {\bf A}_T + {\bf A}_P.\) In other words, when the boundary is at infinity, the two choices of helicity are equivalent. (The winding gauge still exists if there is a finite side boundary, but we will need to kill off a boundary term to establish the equivalence, and in any case we used this boundary condition to get the elegant P-T helicity expression.)
From equation (82) of Prior & Yeates, we have \({\bf A}^W = \mathcal{L}\psi + A_z^W\hat{\bf z},\) where \[ \psi = -\frac{1}{2\pi}\int_{S_z}B_z(y_1,y_2,z)\,\log|{\bf r}|\,\mathrm{d}^2y. \] Now \[ \Delta_\parallel\psi = -\frac{1}{2\pi}\int_{S_z}B_z(y_1,y_2,z)\,\Delta_\parallel\log|{\bf r}|\,\mathrm{d}^2y = -\int_{S_z} B_z(y_1,y_2,z)\delta({\bf r})\,\mathrm{d}^2y = -B_z(x_1, x_2, z). \] This shows that \(P=\psi\) and hence \({\bf A}_P = {\bf A}_h^W.\)
To see that \(A_z^W=T,\) note that \[\begin{align*} A_z^W &= \frac{1}{2\pi}\int_{S_z}\frac{B_1(y_1,y_2,z)r_2 - B_2(y_1,y_2,z)r_1}{|{\bf r}|^2}\,\mathrm{d}^2y\\ &= \frac{1}{2\pi}\int_{S_z}{\bf B}_h(y_1,y_2,z)\times\nabla_h^{[x]}(\log|{\bf r}|)\,\mathrm{d}^2y\\ &= -\frac{1}{2\pi}\int_{S_z}{\bf B}_h(y_1,y_2,z)\times\nabla_h^{[y]}(\log|{\bf r}|)\,\mathrm{d}^2y\\ &= -\frac{1}{2\pi}\int_{S_z}\log|{\bf r}|\nabla^{[y]}\times{\bf B}_h(y_1,y_2,z)\,\mathrm{d}^2y, \end{align*}\] where the last equality uses the fact that the boundary term vanishes at infinity. It follows that \[ \Delta_\parallel A_z^W = -\frac{1}{2\pi}\int_{S_z}j_z(y_1,y_2,z)\Delta_\parallel\log|{\bf r}|\,\mathrm{d}^2y = - \int_{S_z}j_z(y_1,y_2,z)\delta({\bf r})\,\mathrm{d}^2y = -j_z(x_1, x_2, z). \] Therefore \(A_z^W = T\) and so together \({\bf A}^W = {\bf A}_T + {\bf A}_P.\)