Chapter IV: Stochastic Ion Heating in Magnetic Reconnection
4.4 Stochastic Heating Condition Analysis
In the last line, we have used ue ' −∇ ×B so uez = ∂Bx/∂y − ∂By/∂x and
∂By/∂x >> ∂Bx/∂y in the EDR; these assumptions are true becauseBy > Bx and also ∂/∂x >> ∂/∂y. Therefore, B2y/2 acts as an effective electrostatic potential in the x-direction (this is equivalent to Eq. 3 in Li and Horiuchi [93]). This conclusion regarding the effective potential is shown graphically in Fig. 4.1b, where
−∫
Exdxand−∫
uezBydxare compared withB2y/2, calculated from the simulation.
Outside the EDR,∂By/∂x −→ 0, so the approximation fails, but for the purposes of analyzing the slope of the potential valley into which the ions fall (ion inflow directions represented by the green arrows) and thus for the purpose of investigating the existence of stochastic heating, B2y/2 remains a good approximation for the electrostatic potential. The integration constant was set so that the three functions coincide at x= 0 in Fig. 4.1b.
Equation 4.17, the condition for ion stochasticity in the inflow direction, can thus be written as
1 B2y
∂2
∂x2 B2y
2
!
> 1. (4.25)
UsingBy(x) ' B0x/λfor x < λandde2/ε= di2, Eq. 4.25 in dimensioned quantities becomesx2 < di2. A sufficient condition to satisfy x2 < di2whilex < λis
λ2 < di2. (4.26)
Therefore, at sub-ion-skin-depth length scales, at which collisionless reconnection occurs, the condition for stochastic heating in the inflow direction is satisfied.
The cold electron assumption in the inflow direction and thus the validity of Eq.
4.24 is justified by the fact that numerical [93] and experimental [168] studies show that the magnetic force on the electron fluid is balanced by the electric field corresponding to Eq. 4.23 and thus Eq. 4.24.
Outflow
We now consider the outflow (flow along the x = 0 line, magenta dashed line in Fig. 4.1a). Because the magnitude ofBis very small on this line, the cold electron assumption in principle cannot be used. However, it will be shown later that including the electron pressure term has little effect on the stochasticity condition, so for now electrons will be assumed cold. Along the outflow (i.e., on the y-axis, x = 0), the magnetic field is in the x-direction, so the perpendicular electric field relevant to stochastic heating isEy. The magnetic field componentBx at x =0 has
small amplitude and its being finite is what constitutes the reconnection. Ez again plays no role in Eq. 4.1 because∂/∂z = 0. They-component of Eq. 4.19 with the cold electron assumption invoked is
Ey = −∂uey
∂t +uezQex−uexQez− ∂
∂y u2e
2
. (4.27)
Using Eq. 4.20, the first term on the right-hand side of Eq. 4.27 is −∂uey/∂t '
−uey∂uey/∂y =−∂ u2ey/2
/∂y. The second term disappears becauseQex(x =0)= 0 sinceQe does not reconnect. The third term disappears because of the antisym- metry ofuex and the quadrupole nature ofQez. Equation 4.27 now simplifies to
Ey ' − ∂
∂y
u2ey+u2e 2
!
. (4.28)
Thus,
u2ey+u2e
/2 acts as an effective potential in the y-direction. A comparison between −∫
Eydx and
u2ey+u2e
/2 along x = 0 shows that the latter is a good approximation, as shown in Figure 4.1c. The potential forms a hill off of which the ions fall downwards (ion outflow directions represented by green arrows). The integration constant was set in Fig. 4.1c so that the two functions coincide at y= 0.
The normalized stochasticity condition (Eq. 4.17) in the outflow direction is 1
εB2x
∂2
∂y2
u2ey+u2e 2
!
> 1. (4.29)
Here the connected nature ofQegivesQex(x =0)= ∂uez/∂y−Bx 'uez/Ly−Bx = 0, so using∂/∂y∼ 1/Ly = Bx/uezon Eq. 4.29 gives
1 ε
u2ey+u2e 2u2ez
!
> 1. (4.30)
However,
u2ey+u2e
/u2ez ≥ 1 sinceu2e =u2ey+u2ez, so Eq. 4.30 reduces to mi
me
> 1, (4.31)
which is of course satisfied. Consequently, the stochasticity condition in the outflow direction is always satisfied by a wide margin.
The cold electron assumption in the outflow direction will now be justified. Recalling that ¯pe =nekBTe/(B20/µ0)gives ¯pe =n¯ev¯T e2 /2 where ¯vT e2 = (2kBTe/me) /(de|ωce|)2. Dropping bars, the pressure term in Eq. 4.19 becomes ∇pne
e = ∇v2
T e
2
+ vT e22 ∇nnee '
∇v2 T e
2
. The v
2 T e
2
∇ne
ne term was dropped by invoking the quasi-neutrality condition;
because the normalized Poisson’s equation gives ∇ · E = − c2
v2Ae(ne − n0) where v2Ae = B2/µ0mene is the electron Alfvén velocity, taking the gradient of both sides and requiring ∇ ·E to be small by quasi-neutrality yields ∇ne << v2Ae/c2. Using this new pressure term in Eq. 4.19 and applying the same reasoning used to derive Eq. 4.28,Ey approximates to
Ey ' − ∂
∂y
u2ey +u2e+vT e2 2
!
. (4.32)
Equation 4.32 implies that if there is electron heating in the exhaust region, the potential becomes flatter and thus less conducive to stochastic heating. This is not a problem in guide-field reconnection in which electron heating is localized near the EDR [146]. For zero guide-field reconnection, however, if the exhaust region (finite y) is hotter than the EDR (y ≈0) [166], the potential is a well centered aroundy =0 which is not conducive to stochastic heating. Nevertheless, temperature effects do not negate the stochastic condition being satisfied for the following reasons.
Consider the worst case scenario where the outflow becomes completely thermal- ized, i.e., u2ey+vT e2 = const. In this case, onlyu2e is left in Eq. 4.32 so Eq. 4.30 becomes 1ε
u2 e
2u2ez
> 1,which is always satisfied becauseu2e ≥ u2ez.
Furthermore, previous studies are consistent with the cold electron assumption.
Several numerical studies [79, 125] have shown that|eΦ/Te|— which corresponds to ∼ u2e/vT e2 — increases to be 10 ∼ 20 inside the saddle potential, which means that the electron flow is cold and laminar. A previous experimental study [168]
measured the plasma potential including electron temperature effects and showed that the potential is a hill in the y-direction with the peak at y = 0, confirming the shape given in Fig. 4.1(c).
The fact that Eqs. 4.26 and 4.31 do not depend on the magnetic field signifies that stochastic heating is intrinsic to collisionless reconnection. φ in Eq. 4.17 has a dependence on B in such a way that this dependence cancels out B in the denominator. Thus, Eq. 4.17 is always met given that the assumptions in the present analysis are met as well.