Chapter III: Electron Canonical Battery Term: Completion of the Canonical
3.4 Results
Viscosity Dependence
We first examine how electron viscosity affects the evolution of Qe. First recall from Chapter 2 and Eq. 2.27 thatQe ' −Bfor∇−1∼ L 1 whereasQe 'we for L 1. For regions where L ∼ 1 such as the EDR, bothwe andBare important.
Now, electron viscosity contributes to the canonical battery term as
− ∇ ×
∇ · [−µ∇ue] ne
= ∇ ×
µ∇2ue
ne
' µ∇2we
ne
, (3.40)
assuming density variations are not too large. If L 1 so thatQe ' we and thus
∇2we' ∇2Qe, Eq. 2.10 becomes
∂Qe
∂t =∇ × (ue×Qe)+ µ ne
∇2Qe, (3.41)
which has the same form as the resistive-MHD induction equation (Eq. 1.49),
∂B
∂t =∇ × (U×B)+ η
µ0∇2B. (3.42)
Thus at electron scales, electron viscosity allows Qe to reconnect, similar to how resistivity allowsBto reconnect.
Figure 3.3 shows, for three different values of viscosity, the out-of-plane electron flowuez(color), in-planeB(black lines), and in-planeQe(red lines) for a situation with isotropic pressure (initially βe = 0.3). Note that the orientation of the plots is rotated by 90◦ in comparison to those in Chapter 2 to conform to the popular
−4
−2 0 2 4
x/de
(a)µ= 0
−0.9
−0.8−0.7
−0.6
−0.5
−0.4
−0.3−0.2
−0.1 0.0
−4
−2 0 2 4
x/de
(b)µ= 10−6
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1 0.0
−40 −30 −20 −10 0 10 20 30 40 y/de
−4
−2 0 2 4
x/de
(c)µ= 10−5
−0.7−0.6
−0.5
−0.4−0.3
−0.2
−0.1 0.0 0.1
Figure 3.3: In-planeQe(red), in-planeB(black) anduez(color) for varyingµvalues and isotropic pressure att =(a) 380, (b) 450, (c) 560.
orientation in the literature. The shape of the EDR corresponds to the shape of the purple color. Different times are chosen for each viscosity value because viscosity changes how much time is required for the EDR to display its characteristic structure.
Forµ= 0 (Fig. 3.3a) the system is ideal, soQelines remain connected and pile up near thex = 0 line in contrast toBlines which reconnect, as seen in Chapter 2. For finite µ(Figs. 3.3b and 3.3c),Qelines reconnect as well.
It is apparent from Fig.3.3 that the out-of-plane electron current structure (i.e., color contours) is well manifested by in-plane Qe but not by B — a feature that is an important advantage of using Qe over B at electron scales. Fine structures (i.e.
small L) of ue are not manifested byB because |B| ∼ |j|L ∼ |ue|L by Ampére’s law, whereas they are well manifested byQebecause|Qe| ∼ |ue| /L.
Another important feature is that the local increase of Qey shear corresponds to the local increase of uez. To illustrate this point we consider the implications
−4
−2 0 2 4
x/de
(a) Anisotropic,µ= 0
−0.96
−0.84
−0.72
−0.60
−0.48
−0.36
−0.24
−0.12 0.00
−4
−2 0 2 4
x/de
(b) Anisotropic,µ= 10−6
−0.9−0.8
−0.7
−0.6−0.5
−0.4
−0.3−0.2
−0.1 0.0
−40 −30 −20 −10 0 10 20 30 40 y/de
−4
−2 0 2 4
x/de
(c) Anisotropic,µ= 10−5
−0.875
−0.750
−0.625
−0.500
−0.375
−0.250
−0.125 0.000 0.125
Figure 3.4: Same as Fig. 3.3 with pressure anisotropy and varying µvalues att = (a) 400, (b) 500, (c) 630.
of an assumed hypothetical toy scenario where Qey ∼ A(t)xexp −x2
where A(t) increases in time. This profile represents locally sheared Qey. Using uez =
−∫
weydx ' −∫
Qeydxfor L 1 givesuez ∼ A(t)exp −x2
, corresponding to a local increase inuez.
Anisotropy Dependence
We next examine how pressure anisotropy affectsQe. In order to bring the system to an anisotropy-driven state faster, an initial pressure anisotropy with βek = 0.6 and βe⊥ = 0.1 was imposed. The results for different µare shown in Figs. 3.4a- c. In comparison to the isotropic case it is seen that pressure anisotropy greatly distorts the in-planeQe lines so that they pile up in the upper-left and lower-right quadrants; this corresponds to the anisotropic regime in Ohia et al. [111]. Again, the out-of-plane current structure is correlated with Qe rather than with B; uez is enhanced at locations whereQeyis sheared. The distortion ofQefield lines and the
−20 −10 0 10 20 y/de
−2 0 2
x/de
Particle-in-cell
−0.375
−0.300
−0.225
−0.150
−0.075 0.000 0.075 0.150 0.225
Figure 3.5: Same as 3.3 from the particle-in-cell simulation att = 300.
−1 0 1
x/de
0.015 0.030
-0.015
-0.030 0.000
0.000
0.000
(a) ˆy· ∇ ×(ue×Qe) (color),−yˆ· ∇ ×(∇ ·pe,aniso/ne) (contour)
−0.035 0.000 0.035
−30 −20 −10 0 10 20 30
y/de
−1 0 1
x/de
(b)∂Qey/∂t= ˆy· ∇ ×(ue×Qe)−yˆ· ∇ ×(∇ ·pe,aniso/ne)
−0.035 0.000 0.035
Figure 3.6: (a) They-component of the convective term ˆy· ∇ × (ue×Qe)(color) and the anisotropic contribution to the canonical battery term−yˆ· ∇ × ∇ ·pe,aniso/ne (contour) for the simulation corresponding to Fig. 3.4b. (b) The sum of ˆy · ∇ × (ue×Qe)and−yˆ· ∇ × ∇ ·pe,aniso/ne
, which is equal to∂Qey/∂t. The red arrows represent the direction ofQey.
corresponding elongation and tilt of the out-of-plane current are reproduced by the PIC simulation, as shown in Fig. 3.5.
The anisotropic contribution to the electron canonical battery term explains the origin of the distortion of Qe and equivalently of the elongation of uez. Figure 3.6a shows the convective term ˆy · ∇ × (ue×Qe) (color) and the battery term
−yˆ · ∇ × ∇ ·pe,aniso/ne
(contour). Figure 3.6b shows the sum of the two terms which is equal to∂Qey/∂t, and the red arrows show the direction ofQey. It can be seen that−yˆ· ∇ × ∇ ·pe,aniso/ne
adds to ˆy· ∇ × (ue×Qe)and increases the spatial
extent of∂Qey/∂t; this in turn elongates the structure ofuez.
To further understand the distortion of Qe due to anisotropy, we examine the anisotropic contribution to the canonical battery term, which is equal to
−∇ × ∇ ·pe,aniso/ne ' −∇ × (∇ · [σBB])
=−∇ × (B[B· ∇σ]+σB· ∇B). (3.43) Since an increase of Qey shear corresponds to an increase in uez, we consider the y-component of the canonical induction equation (Eq. 3.4) and thus of Eq. 3.43 which is
∂
∂x (B[B· ∇σ]+σB· ∇B)z = ∂
∂x(Bz[B· ∇σ]+σB· ∇Bz). (3.44) Here,σdepends on|B|andBz is of the same order of magnitude asBy. Therefore, since By has the shortest spatial scale of the components of B, it follows that
|∇σ| /σ |∇Bz/|Bz, so
∂
∂x (Bz[B· ∇σ]+σB· ∇Bz) ' ∂(Bz[B· ∇σ])
∂x ' Bz
∂B· ∇σ
∂x . (3.45)
Therefore, we have
−yˆ· ∇ ×
∇ ·pe,aniso ne
' Bz
∂B· ∇σ
∂x (3.46)
Figure 3.7 shows various quantities involved in the calculation ofBz∂[B· ∇σ] /∂x.
The quadrupole out-of-plane Hall fields (plus and minus signs in Fig. 1.5) add and subtract to the background guide field Bz prescribed by Eq. 3.34. This, together with the reduction of |B| due to reconnection, generates a region of low |B| (Fig.
3.7a) with a pronounced tilt. Because pk ∼ n3e/B2 and p⊥ ∼ neB, this |B| tilt generates a tilted region of finite σ (Fig. 3.7b; color). The quantity B · ∇σ, which is the variation of σ along the in-plane B (Fig. 3.7b; lines), is shown in Fig. 3.7c (color), and its gradient ∇ (B· ∇σ) ' xˆ∂[B· ∇σ] /∂x is represented by the black arrows. The resultant −yˆ · ∇ × ∇ ·pe,aniso/ne
' Bz∂[B· ∇σ] /∂x is shown in Fig. 3.7d (contour). The convective term ˆy· ∇ × (ue×Qe)(Fig. 3.7d;
color) enhances Qey shear (Fig. 3.7d; red arrows) as expected from Eq. 2.49, and−yˆ · ∇ × ∇ ·pe,aniso/ne
further enhances this shear in spatial extent in the y direction. The resultant ∂Qey/∂t = yˆ · ∇ × (ue×Qe) − yˆ · ∇ × ∇ ·pe,aniso/ne is shown in Fig. 3.7e; this increase in Qey shear is the origin of the elongated out-of-plane flowuez.
−1 0 1
x/de
(a)|B|
0.882 0.912 0.942 0.972 1.002 1.032 1.062 1.092 1.122 1.152
−1 0 1
x/de
(b)σ(color),B(lines)
0.18 0.21 0.24 0.27 0.30 0.33 0.36 0.39 0.42 0.45
−1 0 1
x/de
(c)B· ∇σ(color),∇(B· ∇σ) (arrows)
−0.0090
−0.0075
−0.0060
−0.0045
−0.0030
−0.0015 0.0000 0.0015 0.0030
−1 0 1
x/de
0.000 -0.015
-0.030 0.015 0.030
(d) ˆy· ∇ ×(ue×Qe) (color),−yˆ· ∇ ×(∇ ·pe,aniso/ne) (contour)
−0.035 0.000 0.035
−30 −20 −10 0 10 20 30
y/de
−1 0 1
x/de
(e) ˆy· ∇ ×(ue×Qe)−yˆ· ∇ ×(∇ ·pe,aniso/ne)
−0.035 0.000 0.035
Figure 3.7: (a-c) Various quantities involved in the calculation of −yˆ · ∇ ×
∇ ·pe,aniso/ne
' Bz∂[B· ∇σ] /∂x for the simulation corresponding to Fig. 3.4b.
(d) The y-component of the convective term ˆy· ∇ × (ue×Qe)and the anisotropic contribution to the canonical battery term −yˆ · ∇ × ∇ ·pe,aniso/ne
, and (e) their sum.