In this section the morphology of the off-diagonal components of the pressure tensor components are shown for the Harris sheet cases. It is also shown how the different contributions in equation (5.7) make up the reconnection electric field in the vicinity of the X-point.
Figure5.13shows filled contour plots of they component of the current density at the time of maximum reconnection rate for each simulation run. The time of maximum reconnection rate corresponds to the peaks of the graphs for each case shown in Figure5.12(b). Figure 5.14are filled contour plots of theycomponent of the electric field at the time of maximum reconnection rate. The electric field in each case has been averaged for2Ω−i 1either side of the time of maximum reconnection rate. This shows how magnetic reconnection generates a strongycomponent of the electric field in the vicinity of the X-Point. The guide fieldsBy0 = 0.0,0.1,0.3,0.5,0.7,0.9,1.0 have been used. In all the figuresx0refers to the approximate position alongxof the position of the X-point. In all cases the position of the X-point alongzis approximately atz= 0.
To demonstrate that the dominant contribution to theEy electric field in the vicinity of the X- point is due to off-diagonal components of the electron pressure tensor Figure5.15and Figure
5.16show line plots alongxandzthrough the X-point of the contributions that make up theEy electric field in Eq. (5.7) at the time of maximum reconnection rate. The quantities in each case have been averaged for2Ω−i1 either side of the time of maximum reconnection rate. Examination of these figures clearly shows that the dominant contribution to the electric field in the vicinity of the X-point comes from the gradients of the off-diagonal components of the electron pressure tensor (green lines).
In each of the plots shown in Figure5.15and Figure5.16the gradients of the off-diagonal com- ponents of the electron pressure tensor (green lines) dominate the electric field out to the point at which theu×Bterm (purple lines) equals the contribution from the gradients of the electron pressure tensor terms. The plots shown in Figure5.15show that the contribution from the off diag- onal pressure tensor components dominates to aboutx≈ ±3c/ωpieither side of the X-point. For the plots shown in Figure5.16the contribution from the off diagonal pressure tensor components
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Figure 5.13: Plots of they component of the current density with contours of the flux function overplotted for simulations with varying strengths of guide field at the time of maximum reconnec- tion rate. (a)By0 = 0.0(x0 =−0.28), (b)By0 = 0.1(x0 =−0.28), (c)By0 = 0.3(x0 = 0.0), (d)By0 = 0.5(x0 = 0.0), (e)By0= 0.7(x0 =−0.28), (f)By0 = 0.9(x0= 0.0), (g)By0 = 1.0 (x0 =−0.28).
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Figure 5.14: Plots of theEyelectric field for simulations with varying strengths of guide field at the time of maximum reconnection rate. (a)By0 = 0.0(x0 =−0.28), (b)By0 = 0.1(x0 =−0.28), (c)By0 = 0.3(x0 = 0.0), (d)By0 = 0.5(x0 = 0.0), (e)By0 = 0.7(x0 =−0.28), (f)By0 = 0.9 (x0 = 0.0), (g)By0 = 1.0(x0=−0.28).
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Figure 5.15: Plots alongxforz= 0of the contributions that make up theEyelectric field in Eq. (5.7) for simulations with varying strengths of guide field at the time of maximum reconnection rate. (a)By0 = 0.0(x0 =−0.28), (b)By0 = 0.1(x0 =−0.28), (c)By0 = 0.3(x0 = 0.0), (d) By0 = 0.5(x0 = 0.0), (e)By0 = 0.7(x0 =−0.28), (f)By0 = 0.9(x0 = 0.0), (g)By0 = 1.0 (x0 =−0.28).
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Figure 5.16: Plots alongzforx=x0of the contributions that make up theEy electric field in Eq. (5.7) for simulations with varying strengths of guide field at the time of maximum reconnection rate. (a)By0 = 0.0(x0 =−0.28), (b)By0 = 0.1(x0 =−0.28), (c)By0 = 0.3(x0 = 0.0), (d) By0 = 0.5(x0 = 0.0), (e)By0 = 0.7(x0 =−0.28), (f)By0 = 0.9(x0 = 0.0), (g)By0 = 1.0 (x0 =−0.28).
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Figure 5.17: Plots of thePxy,e component of the electron pressure tensor for simulations with varying strengths of guide field at the time of maximum reconnection rate. (a)By0 = 0.0(x0 =
−0.28), (b)By0 = 0.1(x0 =−0.28), (c)By0 = 0.3(x0 = 0.0), (d)By0 = 0.5(x0 = 0.0), (e) By0= 0.7(x0 =−0.28), (f)By0 = 0.9(x0 = 0.0), (g)By0= 1.0(x0 =−0.28).
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Figure 5.18: Plots of thePyz,e component of the electron pressure tensor for simulations with varying strengths of guide field at the time of maximum reconnection rate. (a)By0 = 0.0(x0 =
−0.28), (b)By0 = 0.1(x0 =−0.28), (c)By0 = 0.3(x0 = 0.0), (d)By0 = 0.5(x0 = 0.0), (e) By0= 0.7(x0 =−0.28), (f)By0 = 0.9(x0 = 0.0), (g)By0= 1.0(x0 =−0.28).
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Figure 5.19: Plots of the profile ofBz alongx and of the profile of Bx along z going through the position of the X-point at the time of maximum reconnection rate for the zero guide field (anti-parallel) case.
dominates to aboutz≈ ±1.8c/ωpieither side of the X-point for the weak guide field cases whilst for the strong guide field cases they dominate up toz≈ ±0.5c/ωpieither side of the X-point. Although the dominant term in the vicinity of the X-point is due to the gradients of the off- diagonal terms of the electron pressure tensor, there is a small contribution to the electric field from the electron inertia term at the edge of the diffusion regime. It can be seen in the plots along zof the contributions to the reconnection electric field shown in Figure5.16that as the guide field becomes larger the contribution from the electron inertia (black lines) at the edge of the diffusion region becomes larger. In the strong guide field cases this electron inertia term is in fact of equal magnitude to theu×Bterm. In a recent paper byHesse et al.(2004) for a simulation starting from a Harris sheet with a guide field ofBy0 = 0.8, it was shown that within a collisionless skin depth there was a finite contribution from the electron inertia at the edge of the localised current region. The main contribution to theEy electric field close to the X-point was still due to the nongyrotropic pressures. The results shown here are consistent with this. The significant contribution from the electron inertia term is due the small scales associated with the electron Larmor radius in the strong guide field case.
To investigate further the structure of the off-diagonal electron pressure tensor components, Fig- ures5.17and 5.18show filled contour plots of thePxy,e andPyz,e components of the electron pressure tensor zoomed in around the position of the central X-point. In the anti-parallel case thePxy,e component is approximately symmetrical about the line x = x0 which goes through the X-point with a gradient inx. ThePyz,e component of the pressure tensor is approximately
symmetrical about thez = 0line going through the X-point with a gradient inz. As the guide field is increased the structure of thePxy,e component starts to change. The symmetry along the linex =x0is broken and the structure is seen to rotate in an anti-clockwise direction until at the guide fieldBy0= 1.0case thePxy,ecomponent is almost symmetrical along the linez= 0with a gradient inz. The central region in all of the plots can also be seen to thin. ThePyz,ecomponent of the electron pressure tensor is seen to at first rotate in the anti-clockwise direction up to about a guide field ofBy0 = 0.5 but then rotates clockwise until for a guide field ofBy0 = 1.0 the symmetry alongz = 0has almost been restored. The thinning of the central region can again be observed as the guide field becomes stronger.
It is currently unclear why the structure of thePxy,e andPyz,ecomponents of the electron pres- sure tensor have the behaviour shown in Figure5.17and Figure 5.18for the intermediate cases between the zero guide field and strong guide field case. Analytical estimates of thePxy,e and Pyz,ecomponents of the electron pressure tensor in the vicinity of the X-point have been derived for the anti-parallel case (Hesse et al. 1999) and the strong guide field case (Hesse et al. 2004). A full analytical theory that can describe the changing structure ofPxy,eandPyz,eshown in Figure
5.17and Figure5.18as the transition between a zero guide field case to a strong guide field case is made is not currently available and is a problem for future work.
The thinning of the central region in Figure5.17and Figure5.18is due to the introduction of the guide field. In the case with zero guide field the characteristic length scale ofPxy,e andPyz,eis given approximately by the electron bounce widths in a field reversalλx andλz(e.g.Hesse et al.
1999). In these simulations the electron bounce width in thexandzdirections for the anti-parallel field configuration as given by Eqs. (5.3) and (5.4) at the time of maximum reconnection rate are of the orderλx ≈ 2.5c/ωpiandλz ≈ 1.3c/ωpi. Plots of theBxandBz profiles of the magnetic field alongzandxrespectively going through the X-point for the zero guide field case at the time of maximum reconnection rate are shown in Figure5.19. These are used to calculate estimates of the bounce widthsλx andλz. In the strong guide field case the characteristic length scale changes to that of the thermal electron Larmor radiusrL =vth,e/Ωein the guide magnetic field (Hesse et al. 2004). The electron Larmor radius for the simulation withBy0 = 1.0at the time of maximum reconnection rate is of the orderrL ≈ 0.5c/ωpi. This characteristic change of length scale can be seen in Figure5.17and Figure5.18.