The total force, in our experiments, is the sum of the pressure force, the Lorentz force and the viscous force
total force =−∇p+j×B+Fν. (3.3) However, at t = 0tf, the pressure is uniform everywhere and the velocity is zero so the
only force acting on the field is the Lorentz force. This force is presented in Fig. 3.3, for the case with initial uniform current jsep = 1.5, by arrows whose colour and size reflect
the strength of the Lorentz force, and also its direction, in planes across the separator. The Lorentz force will act in towards the separatrix surfaces, which are also plotted in this plane att= 0tf as dashed pale-blue and pink lines, and causes them to fold towards each
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other as the experiment proceeds. Note, in Fig. 3.3b only small parts of the upper nulls separatrix surface (pink lines) intersect this cut forming only two small segments visible near the top and the bottom.
Figure 3.3: Arrows displaying the initial Lorentz force in the planes at (a) z=−0.5, (b)
z=−0.15, (c) z= 0.0, (d) z= 0.5, (e)z= 1.0 and (f) z= 1.5. The arrows indicate the strength (by their size and colour) and direction of the Lorentz force in the plane. The length of the arrows has been normalised to the maximum value of|j×B|in the domain. The pale-blue/pink dashed lines indicate the locations where the initial separatrix surfaces from the lower/upper nulls intersect the given plane, respectively.
Fig. 3.4 displays contours of the total force in the final equilibrium which are drawn in planes perpendicular to the separator. The total force is equal to the Lorentz force plus the pressure force here. The viscous force is negligible since the velocities are small in the equilibrium. The intersections of the lower and upper null’s separatrix surfaces with this cut are plotted here also for the initial magnetic field (dashed lines) and the equilibrium magnetic field (solid lines). The separatrix surfaces have folded towards each other and, particularly at the times plotted in Figs. 3.4c, 3.4d and 3.4e, the separatrix surfaces in the equilibrium field have clearly curved towards each other in agreement with the initial non-zero Lorentz force. Indeed, the two-dimensional cuts, perpendicular to the separator, reveal that the separatrix surfaces form a cusp exactly like that seen in the collapse of the magnetic field about a 2D null point [e.g., Craig and Litvinenko, 2005, Pontin and Craig, 2005, Fuentes-Fern´andez et al., 2011]. The cusp regions form due to the nature of the pressure which is initially uniform but is changed through the relaxation. This is
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discussed in Sect. 3.2.4.
From the contours of the total force we see that the total force is zero everywhere except near the separator and along the separatrix surface of the nearest null to the plane and that the magnitude of these forces are small in comparison to the initial forces shown in Fig. 3.3 (∼4% of the maximum initial Lorentz force). This result is not surprising. A lack of exact force-balance in the local vicinity of topological features is found in the equilibrium field associated with collapsed 2D and 3D null points and an infinite-time collapse is seen [e.g., Klapper, 1998, Craig and Litvinenko, 2005, Fuentes-Fern´andez et al., 2011]. Therefore, the highly-localised, residual forces that we find suggest that separators also undergo an infinite time collapse. This is discussed further in Sect. 3.3.5.
Figure 3.4: Contours of the equilibrium total force in the planes (a) z = −0.5, (b) z =
−0.15, (c) z = 0.0, (d)z = 0.5, (e) z = 1.0 and (f) z = 1.5. The pale-blue/pink dashed lines indicate where the initial separatrix surfaces from the lower/upper nulls intersect the plane. The solid lines represent the positions of these separatrix surfaces once the system has reached its final equilibrium.
In the equilibrium state, the Lorentz force is zero along thez-axis (Figs. 3.5a and 3.5b) and hence along the separator (sincejremains parallel to thez-axis). Therefore, the total force along the separator is made up only of the pressure force, which is plotted along the
z-axis in Figs. 3.5c and 3.5d. The plots in Figs. 3.5e and 3.5f confirm that the total force along the separator itself is indeed made up only of thez-component of the pressure force
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along the z-axis. This force acts outwards towards the nulls from a point just over half way along the separator atz= 0.58.
Figure 3.5: The perpendicular (left column) and parallel (right column) components of the Lorentz force ((a) and (b)), the pressure force ((c) and (d)) and the total force ((e) and (f)) along thez-axis (including the separator) of the equilibrium field.
We have investigated the total force along the length of the current layer, but also want to check its behaviour through the depth and across the width of the current layer. Along these cuts we will analyse the behaviour of various plasma parameters in a number of the following sections. Fig. 3.6 displays a cut perpendicular to the separator in the z = 0.5 plane through the separator current layer in the equilibrium state with filled contours of
|j| (details of this current layer are discussed in Sect. 3.2.5) along with white lines which are plotted through the depth (solid) and across the width (dashed) of the current layer. It is along these lines, in this plane, that we will plot various parameters.
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Fig. 3.7 shows that the Lorentz and pressure forces behave similarly in magnitude, but opposite in sign, through the depth and across the width of the current layer, leading
to the total force almost vanishing here, except as it crosses the current layer. These small residual net forces at the current layer indicate that the current here is still growing, as expected in the case of an infinite-time singularity. Fuentes-Fern´andez et al. [2011] show similar cuts displaying the total force across the width of the current layer formed after the collapse of a 2D null. These plots have the same profile as Fig. 3.7c. Residual forces for the collapse of a 2D null or a 3D separator are therefore found to lie along the current layer.
The velocities in the domain increase sharply from a value of zero at t = 0tf to a
maximum value of |v|= 0.28 throughout the domain at t = 1.28tf. The mean velocity
at this time is ¯|v| = 0.09. The maximum and mean velocities in the system decrease after this time with the maximum value equal to|v|= 0.009 and the mean value equal to
¯
|v|= 0.003 att= 12.82tf. In the equilibrium state the maximum velocity is|v|= 6×10−4
and the mean value is ¯|v|= 9×10−6.
Figure 3.6: Contours of |j| in a cut perpendicular to the separator at z = 0.5 in the equilibrium state. The strong current layer is highlighted here at the centre. Here also, a line is drawn through the depth of the current layer (solid white) and across the width of the current layer (dashed white). The inserted image highlights the depth, d, and width,
w, of the current layer in this plane.