LAS NUEVAS TECNOLOGÍAS Y SU USO ADECUADO EN EL TRABAJO EDUCATIVO DE LAS I.E NUESTRA SEÑORA
LAS NUEVAS TECNOLOGÍAS Y SU USO ADECUADO EN EL TRABAJO EDUCATIVO DE LAS I.E NUESTRA SEÑORA DE LA
2. Marco Conceptual
2.6. Nuevas Tecnologías y Educación
Normalisation Initial Conditions
B0 = 100G = 0.01 Resolution: 1283
L0= 75Mm ρ= 0.1 η0 = 5×10−5
3.2 Investigating Resistivity 60
In Section 3.2.1, the impact of different values of uniform resistivity, when twisting two flux tubes, was considered. We now go on to consider a localised resistivity region of two different forms. The first is of the form:
η=η0e −“r r0 ”2 , (radially dependent) where, r= q (x−0.5)2+ (y−0.5)2+ (z−0.5)2 & r0 = 0.1.
The value of resistivity (η) is exponentially decreasing with radius from the centre of the domain. The second localised resistivity region is of the form:
η=η0e
−“z−r0.5 0
”2
, (z-dependent)
also withr0 = 0.1. This resistivity distribution is constant in thex−y plane and exponentially
decreasing away from the mid-plane. In both of these cases the resistivity is zero at the footpoints of the flux tubes, the location where large currents and heating were found in Section 3.2.1. To examine the effect of a localised resistivity, we compare these forms to the uniform resistivity (η0 = 5×10−5) and an ideal simulation withη = 0(summarised in Table 3.2).
Table 3.2: Localised Resistivity Comparison: Resistivity form and colour scheme that will be used throughout this chapter.
Ideal Uniform Radially dependent z-dependent η= 0 η=η0 η =η0e −“r r0 ”2 η =η0e −“z−0.5 r0 ”2
Plasma Response: Density Evolution
Figure 3.12 shows the variation of the minimum density with time. The uniform resistivity simu- lation, discussed in Section 3.2.1, produced voids of plasma above the footpoints of the flux tubes and has a minimum density (shown in green) of less than10−7. In comparison, for both localised resistivity runs the density decreases slightly over time but remains around 10−3 in normalised units. The two cases behave very similarly to each other, and similarly to the ideal case, as there is no resistivity acting on the footpoints. The large heating and associated reduction in density near the boundaries is therefore not present, as shown in the contours of density att= 40in Fig- ure 3.13. The density distribution in Figure 3.13 is similar for the two localised resistivity cases. In both cases, denser plasma is present towards the central column of the domain, as is also true forη = 0. The density contours for the cases with localised non-ideal regions in Figures 3.13c
3.2 Investigating Resistivity 61
Figure 3.12: Minimum density in the domain with time for simulations with resistivity forms:
η = 0, uniform, radially dependent andz-dependent withη0 = 5×10−5 shown in red, green,
blue and black, respectively.
(a) (b)
(c) (d)
Figure 3.13: Contour plots of density in the domain at t = 40aty = 0.5for simulations with resistivity forms: (a)η = 0, (b) uniform, (c) radially dependent and (d)z-dependent withη0 =
5×10−5.
and 3.13d differ slightly from the ideal case in 3.13a in the centre of the domain. In (c) and (d) there is a slight reduction in density alongx= 0.5visible in the plane. This suggests that Ohmic heating is occurring and thereby reducing the density in this central region whereηis present.
3.2 Investigating Resistivity 62
Energy and Current Evolution
The energy evolution in the domain for the four cases is displayed in Figure 3.14, where the volume integrated energy is plotted separately for the (a) kinetic , (b) internal, (c) magnetic and (d) total energy. As with the uniform resistivity case, the energy evolution is very similar for all forms of resistivity up to approximately t = 20. After this time, the internal energy of the uniform resistivity case (green) begins to increase very quickly in comparison to the other cases and this coincides with the magnetic energy gradient reduction. This is what we would expect as the uniform case has anηpresent in the whole domain, which can act on the high currents near the base to convert the magnetic energy into internal energy earlier. In terms of the magnetic energy evolution, the two localised resistivity regions and the ideal run behave almost identically until
t = 55, when magnetic energy becomes greater for the ideal case. This is because the current formed in the cases with localised non-ideal regions allows the magnetic field to diffuse. The internal energy evolution is also similar for the cases with localised non-ideal regions and the ideal case untilt= 55. After which, the cases with a non zero resistivity produce greater internal energy. This suggests that the localised resistivity does not have a large effect until this time and that diffusion occurring in the uniformηcase is occurring away from these localised regions.
(a) (b)
(c) (d)
Figure 3.14: (a) Kinetic, (b) internal, (c) magnetic and (d) total volume integrated energies for simulations with resistivity forms: η = 0, uniform, radially dependent and z-dependent with
3.2 Investigating Resistivity 63
In Figure 3.14, the case with z-dependent η (black) has slightly larger internal energy than the radially dependent η (blue), but this difference is relatively small (≈ 5%). The magnetic energy evolution is also very similar with minimal difference between them, but thez-dependent
ηdoes have a slightly smaller value at the end of the simulation, which corresponds to the larger internal and kinetic energy. This is due to the larger area where Ohmic heating can occur in thez
dependent case, and so a larger internal energy is produced. The timing of the increase in kinetic energy is also interesting, as the uniformηcase peaks first att≈61in Figure 3.14, but the case with a radially dependent localised non-ideal region increases before the z-dependent η. This suggests that reconnection is occurring earlier in theη = η0e
−(rr
0) 2
case. This initially appears counter-intuitive as the case with a diffusive term acting on a smaller area produces reconnection earlier.
Figure 3.15: The maximum magnitude of current density between 0.46 < z < 0.54 for simulations with resistivity forms: η = 0, uniform, radially dependent and z-dependent with
η0 = 5×10−5 shown in red, green, blue and black, respectively..
(a) (b) (c)
Figure 3.16: Velocity componentvx aty = 0.5in the mid-plane at (a)t = 52, (b)t = 56, (c)
t= 60, for cases with resistivity forms:η = 0, uniform, radially dependent andz-dependent with
3.2 Investigating Resistivity 64
(a) (b)
(c) (d)
Figure 3.17: Contour plots of the current density in the domain aty = 0.5att= 60for simulations with resistivity forms: (a) η = 0, (b) uniform, (c) radially dependent and (d)z-dependent with
η0 = 5×10−5. Over-plotted are the velocity vectors in the mid-plane.
To explain this, we look at how the presence and shape of the localised resistivity region affect the evolution of the field and the timing of the reconnection. As the resistivity regions are both centred around the mid-plane of the domain, they are the same in the centre of the domain but differ at the boundaries. Therefore, we would expect any difference in the behaviour of the two simulations to begin in this area. Figure 3.15 shows the evolution of the maximum current in the narrow region0.46< z <0.54surrounding the mid-plane. Initially theη=η0e
−(rr
0) 2
case (blue) has the same behaviour as the ideal simulation shown in red, with a high maximum current in the mid-plane. In comparison, thez-dependent resistivity has a slightly lower initial maximum current around the mid-plane, before increasing to similar values to the radially dependent case after
t= 35. This initial current disparity between the cases with localised resistivity regions is due to the maximum current near thex-boundaries. Thez-dependent resistivity (black) has a uniformη
in the plane and hence acts to diffuse the field and reduce the currents at thex-boundaries initially. There is no resistivity acting here for the radially dependent localised non-ideal region and so the initial higher currents on the boundaries remain, as with theη = 0case.
The diffusion of the field at thex-boundaries can also affect the magnetic field evolution later in the simulation, through the Lorentz force. When the field has diffused near the boundaries,
3.2 Investigating Resistivity 65
there is a smaller Lorentz force acting towards the centre. The effect of this can be seen in the velocity cut at y = 0.5 in the mid-plane in Figure 3.16. In Figure 3.16a at t = 52, the vx
velocity component has a very similar shape for both runs with localised resistivity regions (in blue and black), where there is a velocity towardsx = 0.5from either side. However, byt= 60
in Figure 3.16c, thevx cut for the radially dependent resistivity region (blue) has a much larger magnitude and steep gradient atx= 0.5. This velocity will act to bring the differently connected field lines together, creating a high current atx = 0.5. This can be seen in Figure 3.17, where contours of the magnitude of the current density (|j|) in the mid-plane are displayed att = 60. The current layer atx= 0.5in contour (c), for the radially dependent resistivity region, is already thinner and stronger than the current layer in contour (d). This is due to the high inwards velocity seen in Figure 3.16 and explains why the maximum current rises earlier (att≈50) for the radially dependent resistivity region in Figure 3.15. This also supports the earlier peak in kinetic energy shown in Figure 3.14, as the current layer is formed earlier and hence allows reconnection to occur earlier.
Evolution of Field line Connectivity
The timing of reconnection can also be inferred from the field line connections traced from one of the sources, these are shown at different times in Figure 3.18. The connectivities in Figure 3.18 are found using the same method outlined in Section 3.2.1 and represent one source onz= 0.0, as it is rotated by the velocity driver and the flux within it is coloured according to its connectivity atz = 1.0. Hence, initially the entire source is dark blue, as all field lines are at their original connection. Then, as they change connectivity, their origin within the source is depicted as light blue. The uniformηcase changes connectivity first and has the largest proportion of the source reconnected at all times. Initially, the two cases with localised resistivity regions show very similar evolution untilt = 60. After this point, the radially dependent form of localisedη has a larger percentage of flux changing connectivity than thez-dependentη. All the cases with some form of resistivity show more flux changing connectivity and at an earlier time than forη = 0, as we would expect as this reconnection is completely numerical.
An estimate of the total percentage of reconnected flux with time is plotted in Figure 3.19. This is estimated using the connections of the field lines traced from both footpoints onz = 0.0
and averaged. The peak percentage of reconnected flux occurs slightly later for thez-dependent
η (black) than for the radially decayingη (blue), which agrees with the timing of reconnection inferred from the energy and current evolutions previously discussed. The peak percentage of reconnected flux for thez-dependentηis also slightly larger, but still less than the uniformηcase. The reconnected flux for all cases (exceptη = 0) peaks within a few time-steps of each other, at approximatelyt= 74, even though they reach different maxima.
3.2 Investigating Resistivity 66 η = 0 η=η0 η =η0e −“r r0 ”2 η=η0e −“z−0.5 r0 ”2
Figure 3.18: The field line connectivity of the right source on z = 0 at different times for simulations with resistivity forms: η = 0, uniform, radially dependent and z-dependent with
η0 = 5×10−5. The solid black line indicates the direction to the centre of rotation. Colours as in
Figure 3.8.
Parallel Electric Field
The condition for reconnection to occur is thatR E||dl6= 0when evaluated on a magnetic field line
passing through a localised region of (one sign of)E||(Schindler et al., 1988). The distribution
of the parallel electric field is displayed in the plane y = 0.5 in Figure 3.20 att = 40, for the three cases with a non-zero resistivity. The parallel electric field is equal to the parallel current multiplied by the resistivity value: ηj||. The uniform resistivity case in Figure 3.20a, shows the
3.2 Investigating Resistivity 67
Figure 3.19: The percentage of flux reconnected from sources for simulations with resistivity forms: η = 0, uniform, radially dependent andz-dependent withη0 = 5×10−5 shown in red,
green, blue and black, respectively.
(a) (b) (c)
Figure 3.20: Contour plots of the parallel electric field (E||) in the domain aty= 0.5att= 40for
simulations with resistivity forms: (a) uniform, (b) radially dependent and (c)z-dependent with
η0 = 5×10−5.
as there is always a high current in all the cases at the footpoints and with a uniform resistivity throughout the domain, it is at this location that the highest value ofE||occurs.
The two localised resistivity values create localised regions of E|| around the centre of the
domain in Figures 3.20b and 3.20c. The peak value occurs in the centre where the highest value of
ηand a relatively large current are located. Both cases have a very similar maximum value ofE||
(at the time shown) in the centre of the domain. The form of the highE||region is slightly broader
for the resistivity that is onlyz-dependent, due to theη not being restricted in thex−y plane, as it is for the radially dependentη shown in Figure 3.20b. Even though the resistivity is present across the plane in thez-dependentηcase, it is only towards the centre of the domain that theE||
occurs, as this is where the current forms from the twisted flux tubes. This distribution ofE|| is
3.2 Investigating Resistivity 68
(a) (b) (c)
Figure 3.21: Field lines coloured with parallel electric field traced from the mid-plane att= 40
(top row) andt= 50(bottom row) for simulations with resistivity forms: (a) uniform, (b) radially dependent and (c)z-dependent withη0 = 5×10−5.
are coloured according to the value ofE||along them. As the field lines in images 3.21b and 3.21c
enter the localised non-ideal region, highE|| values are present along the field lines. Att = 50,
the uniform resistivity case (in Figure 3.21a) shows high values ofE||along longer sections of the
field lines, suggesting reconnection may be occurring in a larger area.
Conservation of Energy
As described in Section 3.2.1, the conservation of energy can be considered by comparing the total rate of change of energy in the domain to the Poynting flux injected atz = 0.0andz= 1.0
(see Equation 3.6). For the uniformη case, as we have previously discussed, these values align throughout the simulation and hence energy is conserved forη0 = 5×10−5. In contrast, theη= 0
case should allow for an ideal evolution of the magnetic field, but numerical reconnection occurs and hence energy conservation is lost. We now consider the impact of a localised resistivity region on the energy conservation.
In Figure 3.22 we consider the energy conservation in the total domain and within subsec- tions centred around the mid-plane. The subsections we consider (illustrated in Figure 3.22e) are
0.025 < z <0.975,0.125< z < 0.875,0.25 < z < 0.75and0.375 < z < 0.625, displayed in dark blue, light blue, green and orange, respectively. When we consider energy conservation in these subsections of the domain, the flux through the planes bounding the regions can be com- pared to the change in the total energy within each of these regions. Contrary to when we consider
3.2 Investigating Resistivity 69
(a) (b)
(c) (d)
(e)
Figure 3.22: Total flux in/out of the domain at cuts0.025< z <0.975(dark blue),0.125< z <
0.875(light blue),0.25 < z < 0.75(green) and 0.375 < z < 0.625(orange) for simulations with resistivity forms: (a)η = 0, (b) uniform, (c) radially dependent and (d)z-dependent with
η0 = 5×10−5. The subsections of the domain we consider are plotted against height in (e).
the whole domain with boundaries on z = 0 andz = 1, when considering the flux into these smaller domains, there is a vertical velocity through the plane to take into account. Therefore, as described in Section 3.2.1, the kinetic energy flux and enthalpy flux must also be included as well as the Poynting flux in Equation 3.8. The crosses in Figure 3.22 now depict the total flux (Poynting flux + kinetic energy flux + enthalpy flux) into each of these regions, while the solid line shows the rate of change of total energy in the subregion (calculated from the output at each time-step of the code). In all four simulations, approximately25%of the Poynting flux into the
3.2 Investigating Resistivity 70
whole domain (black crosses) reaches the central (orange) subsection.
At all cuts in the domain, the uniformηcase shows energy conservation, as the flux and rate of change of energy align very well. In comparison, for the localised η cases in Figures 3.22c and Figures 3.22d, there is some loss of energy conservation in the whole domain (shown by the difference between the black crosses and the black solid line). However, in the smaller subsec- tions towards the centre of the domain, where the resistivity is present, the energy conservation improves for the localisedηcases. This suggests that the loss of energy conservation is occurring near the footpoints whereη is zero. Both the cases with localised non-ideal regions, show better energy conservation in all subsections of the domain compared to the ideal case in Figure 3.22a. This is most evident in the smallest central section (orange), which shows substantially better conservation for both the localisedηcases in (c) and (d), compared to the idealη = 0case in (a). Finally, considering the domain as a whole, Figure 3.22 shows that the radially dependentη
has a maximum loss of energy conservation of 13% (at t ≈ 62), compared to ≈ 20% for the
z-dependentη (at t ≈ 66). Thez-dependentη, which had the higher maximum current value in Figure 3.15, therefore has the larger loss of energy conservation between the two cases. For both cases, the timing of the maximum loss in energy conservation also coincides with the timing of the initial maximum current (in Figure 3.15) and the peak percentage of reconnected flux in Figure 3.19. The earlier loss of energy conservation therefore supports the previous assertion that the radially dependent ηevolves earlier than the z-dependent case. We also note that these comparisons are carried out with low resolution and therefore the percentages of energy loss are relatively high (see Chapter 4).
Summary and Discussion
In this section, we have compared and analysed the effect of localised resistivity regions in our simulations and compared them with the uniform η case, discussed in Section 3.2.1. We have described two localised forms of η (z-dependent and radially dependent) and considered how their distributions affect the evolution of the reconnecting flux tubes.
We have found that some numerical diffusion occurs for both the localisedη cases, as there is no η at the footpoints and hence (when considering the full domain) energy conservation is lost. However, in the central region where the η is prescribed, the energy conservation is good for the vast majority of the length of the simulations. Looking at the energy evolution and field line connections, reconnection occurs sooner for the uniformηcase. However, we also found that