(a) Perpendicular energy spectrum for runs A (solid), B (dot), C (dash), and D (dash-dot). Note how the spectrum becomes steeper in the HMHD simulations for wavenumbers larger than the inverse ion skin depth (respectively, 32, 16, and 8 for runs B, C, and D). The slope indicates Kolmogorov scaling as a reference. (b) Perpendicular energy spectrum compensated by for the same runs.
Axisymmetric structure functions for the longitudinal magnetic field up to six order for (a) run A ( ), and (b) run D ( ). The order of the structure function is indicated as follows: p = 1 (solid), 2 (dot), 3 (dash), 4 (dash-dot), 5 (dash-triple-dot), and 6 (long dash).
Scaling exponents (with error bars) as a function of the order p up to sixth order, for the velocity (crosses), and for the magnetic field (diamonds), (a) for run A ( ), and (b) for run D ( ). Linear scaling of the exponents with (corresponding to non-intermittent scaling with the second order exponent consistent with the scaling of the energy spectrum in Fig. 1 ) is indicated in both cases by the straight line.
Fourth order structure function of longitudinal magnetic field increments for runs A ( , solid line), B ( , dashed line), C ( , dashed-dotted line), and D ( , dotted line).
PDFs for magnetic field increments, for l = 1.6 (solid), 0.8 (dot), 0.4 (dash-dot), 0.2 (dash-triple-dot), and 0.1 (long dashes), and for runs A, B, C, and D from top to bottom, respectively. In all the figures, a dashed curve indicates a Gaussian PDF with unit variance.
PDFs for velocity field increments, for l = 1.6 (solid), 0.8 (dot), 0.4 (dash-dot), 0.2 (dash-triple-dot), and 0.1 (long dashes), and for runs A, B, C, and D from top to bottom, respectively. In all the figures, a dashed curve indicates a Gaussian PDF with unit variance.
(a) Velocity field scaling exponents (with error bars) as a function of the order p up to sixth order, for runs D (stars), D2 (crosses), and D3 (diamonds) all with . Linear scaling of the exponents is indicated as a reference. (b) Same for the magnetic field scaling exponents.
(a) PDFs of velocity field increments for l = 0.1 and , for runs D (solid line), D2 (dotted line), and D3 (dashed-dotted line). The three PDFs are practically indistinguishable. The dashed line shows a Gaussian distribution as a reference. (b) Same for magnetic field increments.
Skewness (S) and kurtosis (K) for the x-derivatives of bx and ux , for all runs with spatial resolution and with different amplitudes of the Hall effect ϵ. and are, respectively, the skewness and kurtosis of the magnetic field spatial derivatives, while and are the corresponding quantities for the velocity field derivatives.
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