Extended Data Fig. 3: Spectrum of nonlinear energy transfer to the mean magnetic field via turbulent field-line stretching.

This figure confirms that the large-scale velocity fluctuations give energy to the mean magnetic field (red)—and that the small-scale velocity fluctuations receive energy from the mean magnetic field (blue). The quantity shown is T(k″), which represents the rate of nonlinear energy transfer to the mean field \(\hat{b}\)_x(k_x = 0, k_y = 0). Mathematically, T(k″) = ⟨\(\hat{b}\)_x^*(0,0) [\(\hat{{\bf{b}}}\)(k′) · ∇\(\hat{u}\)_x(k″)]⟩_z, where k″ = (k_x″,k_y″) is the horizontal wavenumber of the velocity u, and k′ = (k_x′,k_y′) is the horizontal wavenumber of the magnetic field b; the wavenumber-triad constraint imposes (0,0) = k′ + k″. The quantities \(\hat{{\bf{b}}}\)(k′) and \(\hat{u}\)_x(k″) represent the horizontally Fourier-transformed coefficients of magnetic fields and velocity with wavenumbers k′ and k″, respectively; these quantities are retained in the physical domain z before integrating the transfer function along z. The quantity \(\hat{b}\)_x^*(0,0) is the horizontally-Fourier-transformed, complex-conjugated, x-directed mean magnetic field, whose energy is around two orders of magnitude larger than the y-directed mean field in the nonlinear phase. The mean field \(\hat{b}\)_x(0,0) is inhomogeneous in z (as is the mean flow); the operation ⟨·⟩_z averages the nonlinear transfer function in z. The transfer function is time-averaged over the saturated phase (t = 1000–4000). Of particular note is the extraordinary contribution of the large-scale jets, especially with wavenumber (k_x″a,k_y″a) = (0,0.2), which contribute the largest in the mean-field generation. This is consistent with Fig. 3. The square box at k_x″ = k_y″ = 0 is white because the mean flow does not directly couple to the mean field in this system (the so-called \(\varOmega \)-effect³² is zero).