tokens: analysis to a three-dimensional (3D) turbulent dynamo with periodic boundary conditions and an equator in the middle that cuts the domain into two halves, with opposite helicity of the forcing function. Finally, we employ full-disk vector magnetograms from the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory ( SDO ) to obtain all three magnetic field components at the two-dimensional (2D) surface of the Sun, to compute magnetic helicity spectra from a sequence of synoptic maps. 2. Formalism 2.1. The Usual Magnetic Energy and Helicity Spectra In a periodic Cartesian domain, the usual magnetic energy spectrum is given by the integral over shells in wavenumber space of the Fourier-transformed magnetic energy density,where is the surface differential in Fourier space in D dimensions. In D = 3 dimensions, the surface of a D -dimensional unit sphere is . In D = 2 dimensions, is the circumference of a unit circle, while in D = 1 dimensions, corresponds to the two end points of a line. Here and in the following discussion, we measure the magnetic energy density in rather than , so the vacuum permeability factor is dropped. Furthermore, is the radius of a sphere in Fourier space, and hats denote the Fourier transform of the magnetic field, that is,The magnetic helicity spectrum is defined analogously to aswhere is the Fourier transform of the vector potential , with and . These spectra are normalized such that where angle brackets with subscript V denote volume averages. Analogously, one can define the current helicity spectrum such that , where is proportional to the current density. For the following, it is useful to remember that the magnetic energy spectrum is the Fourier transform of the trace of the two-point correlation tensorwhere angle brackets denote ensemble averaging, which could be approximated by averaging over time, and is assumed to be statistically independent of , owing to the assumption of homogeneity. Thus, following standard relationships (e.g., Matthaeus et al. 1982 ; Brandenburg et al. 2011 ), the energy spectrum is then given by , where is the Fourier transform of M ij . Likewise, the scaled magnetic helicity spectrum is given by , where is the unit vector of . (The hat on is not to be confused with the hats on or M ij , where they denote Fourier transforms over .) (Including here the factors 2 and k on the left sides of the two equations has the advantage that and thus have the same prefactors.) Note that under isotropic conditions, . Such spectra show the high-wavenumber range better than just or , and are therefore also considered in some of the following cases. The magnetic helicity spectrum can be defined analogously from a tensorial generalization of the magnetic two-point correlation tensor. This will be done in the following section, where we will also relax the assumption of homogeneity and allow the spectra to be slowly varying functions of . 2.2. Double Fourier Transform To obtain
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