Cancellation properties in Hall magnetohydrodynamics with a strong guide magnetic field

We present a signed measure analysis of compressible Hall magnetohydrodynamic turbulence with an external guide field. Signed measure analysis allows us to characterize the scaling behavior of the sign-oscillating flow structures and their geometrical properties (fractal dimensions of structures). A reduced numerical model, valid when a strong guide magnetic field is present, is used here. In order to discuss the effect of the Hall term, different values for the ion skin depth are considered in the simulations. Results show that as the Hall term is increased, the fractal dimension of the current and vorticity sheets decreases. This observation, together with previous analysis of the same fields, provides a comprehensive description of the effect of the Hall force on the formation of structures. Two main processes are identified, namely, the widening and unraveling of the sheets.

Figure 1

The left panels show slices of the perpendicular current component $j_x$ (in adimensional units; see Sec. 2 for the normalization used) in the perpendicular plane for the four different values of $\varepsilon$ used in this work ($\varepsilon$ increasing from top to bottom). The right panels show the same fields, seen at an arbitrary tilt angle, highlighting the presence of alternate sign structures at all scales. In these panels, the scale of grays is arbitrary.

Figure 2

The left panels show slices of the parallel current component $j_z$ (in adimensional units; see Sec. 2 for the normalization used) in the perpendicular plane for the four different values of $\varepsilon$ used in this work ($\varepsilon$ increasing from top to bottom). The right panels show the same fields, seen at an arbitrary tilt angle (arbitrary grayscale).

Figure 3

Total energy spectra for the four runs (see the legend). The vertical lines represents the values of the inverse ion skin depth for the three runs with nonvanishing Hall term. Phenomenological predictions for the MHD range and the Hall range are also indicated.

Figure 4

Signed measure $\mu$ as estimated for $j_x$ in the $y$-$z$ plane, shown for run 3 ($\varepsilon =1/16$), for four different partition box sizes (top left, $l_{\perp }=0.12$; top right, $l_{\perp }=0.04$; bottom left, $l_{\perp }=0.016$; and bottom right, $l_{\perp }=0.002$.

Figure 5

Cancellation function $\chi \left(l_{\perp }\right)$ versus the scale parameter $l_{\perp }$. The examples given here refer to the current perpendicular component $j_x$ (left panel) and parallel component $j_z$ (right panel) for $\varepsilon =1/16$.

Figure 6

Cancellation function $\chi \left(l_{\perp }\right)$ for the current perpendicular component $j_x$ (left panel) and parallel component $j_z$ (right panel) for four values of the Hall parameter $\varepsilon$ (see the legend) at $l_{\parallel }/2\pi =0.03$.

Figure 7

Examples of fit of the cancellation function, shown here at $\varepsilon =0$ for $j_z$ (left panel) and at $\varepsilon =1/8$ for $j_x$ (right panel). The power law fits $\chi \left(l_{\perp }\right)=A{(l_{\perp }/2\pi )}^{-\kappa }$ are superimposed (one in the left panel, two in the right panel). The ion skin depth is indicated by the dashed line in the right panel. The bottom part of the right plot shows the compensated cancellation functions $\chi \left(l_{\perp }\right)/A{(l_{\perp }/2\pi )}^{-\kappa }$.

Figure 8

Fractal dimension $D_{\perp }$ estimated through Eq. (18) for the three components of current [for (a) the MHD range and (c) the Hall range] and vorticity [for (b) the MHD range and (d) the Hall range], labeled with different colors and line style (see the inset). The indicators $D_{\perp }^{☆}$ (see the text) are also plotted for (e) and (f) the two fields (black lines).

Figure 9

Cancellation function $\chi \left(l_{\parallel }\right)$ for $j_x$ at $\varepsilon =1/16$ and for $l_{\perp }/2\pi =0.002$. A power law fit is superimposed. The bottom part of the plot shows the compensated cancellation function $\chi \left(l_{\parallel }\right)/A{(l_{\parallel }/2\pi )}^{-\kappa }$.

Figure 10

Fractal dimension $D$ of the parallel cancellation function for the three components of (a) and (c) current and (b) and (d) vorticity. The overall indicators $D_{\parallel }^{☆}$ are also plotted for (c) and (d) the two fields.