tokens: gradually distorted until a singular point appears, where the derivative is discontinuous. Further increasing y , the singular point moves at different x positions, thus generating a discontinuity that is also in the derivative . Starting from the position y = 0 where the solution is regular, the singularity develops over a distance ( y ∼ 6), which is of the order of the typical wavelength. This singularity is found also for other values of z , indicating that a singular surface is present in the 3D structure of A z . At the singular surface the components B x and B y are discontinuous, therefore the surface represent a current sheet. In contrast, the component // is regular in the whole domain. In Figure 1 (bottom panel) the profiles of the magnetic field components and of are plotted as functions of x , for y = 6.13 and z = 0.78. The presence of the discontinuity is clearly visible, mainly in the B y profile. The profile of is remarkably constant, except for a dip localized at the singular point. In fact, the constancy of is guaranteed by Equation ( 1 ), while the smoothing procedure, which modifies the “ideal” solution mostly around the singular point, represents an additional element that does not allow the solution A z to exactly satisfy Equation ( 1 ). Figure 1. Case A. Top panel: profiles of as functions of x , for various values of y , and at z = 0.78. Bottom panel: profiles of B x , B y , B z , and as functions of x , at y = 6.13 and z = 0.78. To study a situation that is closer to the solar wind turbulence, in Case B we considered more complex forms for A x ( x , y , z ), A y ( x , y , z ), and a z ( x , z ) that have 3D power-law spectra (2D for a z ). For the evaluation of these quantities, we used a synthetic turbulence model (STM) introduced by Malara et al. ( 2016 ) and already employed in modeling magnetic turbulence (Pucci et al. 2016 ; Perri et al. 2017 ). The STM builds up a turbulent field as a superposition of localized fluctuations with different amplitudes at different logarithmically distributed spatial scales. The model can reproduce an anisotropic spectrum (Malara et al. 2016 ) that follows the critical-balance scaling law (Goldreich Sridhar 1995 ), where and l ⊥ are the perturbation scale lengths parallel and perpendicular to the mean magnetic field , respectively. Using the STM, we generated A x , A y and the boundary condition a z in the scale ranges π ≤ l ⊥ ≤ π /32, . On average, the amplitude of a fluctuation in the spectrum of the vector potential i th component scales as . For h = 1/3 the spectrum of the spatial derivatives of has a Kolmogorov index () in the perpendicular directions, while in the parallel direction. We choose the parallel direction along x , which is approximately the direction of (see below). The STM can also reproduce intermittency through a p -model technique (Meneveau Sreenivasan 1987 ); in the present case we used p = 0.5, corresponding to no intermittency. A more detailed description of
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