Ideal quadratic invariance In the late 1950s,
Lodewijk Woltjer and
Walter M. Elsässer discovered independently the
ideal invariance of magnetic helicity, that is, its conservation when resistivity is zero. The following outlines Woltjer's proof for a closed system. In
ideal magnetohydrodynamics, the time evolution of a magnetic field and magnetic vector potential can be expressed using the
induction equation as \frac{\partial {\mathbf B}}{\partial t} = \nabla \times ({\mathbf v} \times {\mathbf B}),\quad \frac{\partial {\mathbf A}}{\partial t} = {\mathbf v} \times {\mathbf B} + \nabla\Phi, respectively, where \nabla\Phi is a
scalar potential given by the gauge condition, see
Gauge considerations. Choosing the gauge so that the scalar potential vanishes, \nabla \Phi = \mathbf{0}, the time evolution of magnetic helicity in a volume V is given by: \begin{align} \frac{\partial H^{\mathbf M}}{\partial t} &= \int_V \left( \frac{\partial {\mathbf A}}{\partial t} \cdot {\mathbf B} + {\mathbf A} \cdot \frac{\partial {\mathbf B}}{\partial t} \right) dV \\ &= \int_V ({\mathbf v} \times {\mathbf B}) \cdot{\mathbf B}\ dV + \int_V {\mathbf A} \cdot \left(\nabla \times \frac{\partial {\mathbf A}}{\partial t}\right) dV . \end{align} The
dot product in the integrand of the first term is zero since {\mathbf B} is orthogonal to the cross product {\mathbf v} \times {\mathbf B}. The second term can be integrated by parts to give \frac{\partial H^{\mathbf M}}{\partial t} = \int_V \left(\nabla \times {\mathbf A}\right) \cdot \frac{\partial {\mathbf A}}{\partial t}\ dV + \int_{\partial V} \left({\mathbf A} \times \frac{\partial {\mathbf A}}{\partial t}\right) \cdot d\mathbf{S} where the second term is a
surface integral over the boundary surface \partial V of the closed system. The dot product in the integrand of the first term is zero because \nabla \times {\mathbf A} = {\mathbf B} is orthogonal to \partial {\mathbf A}/\partial t . The second term also vanishes because motions inside the closed system do not affect the vector potential outside, so that at the boundary surface \partial {\mathbf A}/\partial t = \mathbf{0} since the magnetic vector potential is a continuous function. Therefore, \frac{\partial H^{\mathbf M}}{\partial t} = 0 , and magnetic helicity is ideally conserved. In all situations where magnetic helicity is gauge invariant, magnetic helicity is ideally conserved without the need for the specific gauge choice \nabla \Phi = \mathbf{0} . Magnetic helicity remains conserved to a good approximation even with small but finite resistivity. In that case
magnetic reconnection dissipates
energy. As a consequence, the presence of magnetic helicity is a candidate explanation for the existence and sustainment of large-scale magnetic structures in the Universe. The following argument for inverse transfer follows Frisch et al. It is based on the "realizability condition" for the magnetic helicity Fourier spectrum \hat{H}^M_{\mathbf k} = \hat{\mathbf A}^*_{\mathbf k} \cdot \hat{\mathbf B}_{\mathbf k} where \hat{\mathbf B}_{\mathbf k} is the Fourier coefficient at the
wavevector {\mathbf k} of the magnetic field {\mathbf B} , and similarly for \hat{\mathbf A} , the star denoting the
complex conjugate. The realizability condition is an application of the
Cauchy–Schwarz inequality and yields \left|\hat{H}^M_{\mathbf k}\right| \leq \frac{2E^M_{\mathbf k}} , with E^M_{\mathbf k} = \frac{1}{2} \hat{\mathbf B}^*_{\mathbf k}\cdot\hat{\mathbf B}_{\mathbf k} the magnetic energy spectrum. To obtain this inequality, use the relation |\hat{\mathbf B}_{\mathbf k}|=|{\mathbf k}||\hat{\mathbf A}^\perp_{\mathbf k}| , with \hat{\mathbf A}^\perp_{\mathbf k} the
solenoidal part of the Fourier transformed magnetic vector potential orthogonal to the wavevector, since \hat{\mathbf{B}}_{\mathbf k} = i {\mathbf k} \times \hat{\mathbf{A}}_{\mathbf k} . The factor 2 is not present in Frisch et al. because magnetic helicity is defined there as \frac{1}{2} \int_V {\mathbf A} \cdot {\mathbf B}\ dV . Consider an initial state with no velocity field and a magnetic field present only at two wavevectors \mathbf p and \mathbf q . Assume a fully helical magnetic field that saturates the realizability condition, \left|\hat{H}^M_{\mathbf p}\right| = \frac{2E^M_{\mathbf p}} and \left|\hat{H}^M_{\mathbf q}\right| = \frac{2E^M_{\mathbf q}} . If all the energy and magnetic helicity transfer to another wavevector \mathbf k , conservation of magnetic helicity and of the total energy E^T = E^M + E^K , the sum of magnetic and kinetic energy, gives H^M_{\mathbf k} = H^M_{\mathbf p} + H^M_{\mathbf q}, E^T_{\mathbf k} = E^T_{\mathbf p}+E^T_{\mathbf q} = E^M_{\mathbf p}+E^M_{\mathbf q}. Because the initial state has no kinetic energy, it follows that |\mathbf k| \leq \max(|\mathbf p|, |\mathbf q| ) . If instead |\mathbf k| > \max(|\mathbf p|,|\mathbf q| ) , then H^M_{\mathbf k} = H^M_{\mathbf p} + H^M_{\mathbf q} = \frac{2E^M_{\mathbf p}} + \frac{2E^M_{\mathbf q}} > \frac{2\left(E^M_{\mathbf p} + E^M_{\mathbf q}\right)} = \frac{2E^T_{\mathbf k}} \geq \frac{2E^M_{\mathbf k}}, which would violate the realizability condition. Therefore |\mathbf k| \leq \max(|\mathbf p|,|\mathbf q| ) . In particular, for |{\mathbf p}| = |{\mathbf q}| , the magnetic helicity is transferred to a smaller wavevector, which corresponds to larger spatial scales. == See also ==