Sweet-Parker Reconnection

Figure: 9.9 Equally-spaced contours of the magnetic flux-function, $\hat{\psi}(\hat{x},\xi)$, in the vicinity of a non-constant-$\psi$ magnetic island chain. The thick horizontal lines indicate the locations of the Sweet-Parker current sheets.

We saw, in the last section, how a constant-$\psi$ tearing mode evolves in time after it enters the nonlinear regime. Let us now consider how a non-constant-$\psi$ tearing mode evolves in time after it enters the nonlinear regime. Recall, from Section 9.6, that a linear non-constant-$\psi$ tearing mode has a normalized layer thickness $\hat{\delta}\sim S^{-1/3}$, a growth-rate $\gamma\sim S^{-1/3}/\tau_H=S^{2/3}/\tau_R$, and is characterized by ${\mit\Delta}'\,S^{-1/3}\sim 1$. Moreover, according to Section 9.7, the mode enters the nonlinear regime as soon as $\hat{\mit\Psi}^{1/2}$ exceeds $\hat{\delta}$, which implies that ${\mit\Delta}' \,\hat{\mit\Psi}^{1/2}\geq 1$ in the nonlinear regime. Hence, Equations (9.84) and (9.106) lead to the conclusion that $\hat{J}\geq 1$ in the nonlinear regime. In other words, the perturbed current density in the island region exceeds the equilibrium current density. Under these circumstances, both analytical calculations (Waelbroeck 1989) and numerical simulations (Biskamp 1993; Fitzpatrick 2003) suggest that the solution to Equation (9.91) takes the form shown schematically in Figure 9.9. It can be seen that the non-constant-$\psi$ magnetic islands only occupy the regions of the $\xi$-axis in which $\cos\xi<0$, and are connected by thin (compared to both the equilibrium current sheet thickness and the island width) current sheets that run along the resonant surface ($\hat{x}=0$), and occupy the regions of the $\xi$-axis in which $\cos\xi>0$.

Unfortunately, there is no known analytic solution of Equations (9.89)–(9.92) in the non-constant-$\psi$ limit. However, we can still estimate the rate of magnetic reconnection using the so-called the Sweet-Parker model (Sweet 1958; Parker 1957). The Sweet-Parker model concentrates on the dynamics of the current sheets that connect the magnetic islands. The main features of the envisioned magnetic and plasma flow fields in the vicinity of a given current sheet are illustrated in Figure 9.10. The reconnecting magnetic fields are anti-parallel, and of equal strength, $B_\ast$. The current sheet forms at the boundary between the two fields, where the direction of the magnetic field suddenly changes, and is assumed to be of thickness $\delta_\ast$ (in the $x$-direction), and of length $L$ (in the $y$-direction).

Figure 9.10: A Sweet-Parker current sheet.

Plasma is assumed to diffuse into the current sheet, along its whole length, at some relatively small inflow velocity, $v_0$. The plasma is accelerated along the sheet, and eventually expelled from its two ends at some relatively large exit velocity, $v_\ast$. The inflow velocity is simply an ${\bf E}\times{\bf B}$ velocity, so

$$v_0 \sim \frac{E_z}{B_\ast}.$$ (9.138)

The $z$-component of Ohm's law yields

$$E_z \sim \frac{\eta\,B_\ast}{\mu_0\,\delta_\ast}.$$ (9.139)

Continuity of plasma flow inside the sheet gives

$$L\,v_0 \sim \delta_\ast\,v_\ast,$$ (9.140)

assuming incompressible flow. Pressure balance along the length of the sheet yields

$$\frac{B_\ast^{\,2}}{\mu_0} \sim \rho\,v_\ast^{\,2}.$$ (9.141)

Here, we have balanced the magnetic pressure at the center of the sheet against the dynamic pressure of the outflowing plasma at the ends of the sheet. Finally, we can characterize the rate of reconnection via the inflow velocity, $v_0$, because all of the magnetic field-lines that are convected into the sheet, along with the plasma, are eventually reconnected. In fact,

$$\frac{d{\mit\Psi}}{dt} \sim v_0\,B_\ast,$$ (9.142)

where ${\mit\Psi}$ is the (unnormalized) reconnected magnetic flux.

Let us adopt our standard normalizations: $B_\ast=B_0\,\hat{B}_\ast$, ${\mit\Psi} = a\,B_0\,\hat{\mit\Psi}$, $t=\tau_R\,\hat{t}$, $\delta_\ast=a\,\hat{\delta}_\ast$, $v_0= (a/\tau_R)\,\hat{v}_0$, $v_\ast = (a/\tau_R)\,\hat{v}_\ast$, where $\tau_R$ is specified in Equation (9.25). Equations (9.138) and (9.139) yield

$$\hat{v}_0 \sim \frac{1}{\hat{\delta}_\ast}.$$ (9.143)

Equation (9.141) gives

$$\hat{v}_\ast\sim \frac{\hat{B}_\ast\,S}{\hat{k}},$$ (9.144)

where $S=\tau_R/\tau_H$, $\hat{k}=k\,a$, and $\tau_H$ is specified in Equation (9.24). The previous equation suggests that plasma is ejected from the ends of the current sheet at a velocity that is comparable with the Alfvén velocity, $V_A = B_\ast/\!\sqrt{\mu_0\,\rho}$. The length of a given Sweet-Parker current sheet is $L=\pi/k$ (recall that $\xi=k\,y +\varphi$, where $\varphi$ is a constant). Hence, the continuity equation (9.140) implies that

$$\hat{\delta}_\ast \sim\frac{\pi}{\hat{k}}\,\frac{\hat{v}_0}{\hat{v}_\ast}.$$ (9.145)

The previous three equations can be combined to give

$$\hat{\delta}_\ast \sim \left(\frac{\pi}{\hat{B}_\ast\,S}\right)^{1/2},$$ (9.146)

$$\hat{v}_0 \sim \left(\frac{\hat{B}_\ast\,S}{\pi}\right)^{1/2}.$$ (9.147)

It is clear from Equation (9.146) that a Sweet-Parker current sheet is very thin. In fact, the current sheet thickness is of order $S^{1/2}$ times smaller than the equilibrium current sheet thickness. Finally, the normalized version of Equation (9.142) yields

$$\frac{d\hat{\mit\Psi}}{d\hat{t}} \sim \hat{v}_0\,\hat{B}_\ast = \frac{\hat{B}_\ast^{\,3/2}\,S^{1/2}}{\sqrt{\pi}}.$$ (9.148)

Let us make the estimate

$$\hat{B}_\ast = \frac{B_{0\,y}(W/2)}{B_0},$$ (9.149)

where the equilibrium magnetic field, ${\bf B}_0$, is specified in Equation (9.16), and $W=4\,\hat{\mit\Psi}^{1/2}\,a$ is the full width of the magnetic island chain. [Here, we are assuming that (9.84) also applies to non-constant-$\psi$ island chains.] Thus, in the small island width limit, $W\ll a$, we get

$$\hat{B}_\ast = 2\,\hat{\mit\Psi}^{1/2}.$$ (9.150)

Equations (9.148) and (9.150) can be combined to give

$$\frac{d\hat{\mit\Psi}}{d\hat{t}}= \frac{2^{3/2}}{\sqrt{\pi}}\,S^{1/2}\,\hat{\mit\Psi}^{3/4}.$$ (9.151)

It follows that

$$\hat{\mit\Psi}(t) = \left(\frac{t}{\tau_{\rm SP}}\right)^{4},$$ (9.152)

assuming that $\hat{\mit\Psi}(0) = 0$, where

$$\tau_{\rm SP} = \sqrt{2\pi}\,\tau_H^{1/2}\,\tau_R^{1/2}.$$ (9.153)

According to Equation (9.152), once a non-constant-$\psi$ tearing mode enters the nonlinear regime, complete magnetic reconnection (i.e., $\hat{\mit\Psi}\sim 1$) is achieved on the timescale $\tau_{\rm SP}$. Note that this timescale is considerably shorter than the timescale ($\tau_R$) needed for a nonlinear constant-$\psi$ tearing mode to achieve full reconnection (because $\tau_R\gg \tau_H$ in a high Lundquist number plasma).

Equation (9.151) can also be written in the form

$$\frac{d\ln{\mit\Psi}}{d\hat{t}} = \frac{2^{3/2}}{\sqrt{\pi}}\left(\frac{\hat{\delta}}{\hat{\mit\Psi}^{1/2}}\right)^{1/2} S^{2/3}$$ (9.154)

where $\hat{\delta}= S^{-1/3}$ is the non-constant-$\psi$ linear layer thickness. The previous equation confirms that the normalized Sweet-Parker growth-rate matches the normalized non-constant-$\psi$ linear growth-rate (i.e., $S^{\,2/3}$) at the boundary between the linear and nonlinear regimes (i.e., $\hat{\mit\Psi}^{1/2}\sim \hat{\delta}$). Note that the reconnection rate decelerates slightly as the mode enters the nonlinear regime.

A nonlinear constant-$\psi$ tearing mode grows on the very long resistive timescale, $\tau_R$, because plasma inertia plays no role in the reconnection process. This is true despite the existence of an inertial layer on the magnetic separatrix. (See Section 9.8.) On the other hand, a nonlinear non-constant-$\psi$ tearing mode grows on the much shorter hybrid timescale $\tau_H^{1/2}\,\tau_R^{1/2}$ because plasma inertia is able to play a significant role in the reconnection process within the Sweet-Parker current sheets that connect the magnetic islands (but remains negligible outside the sheets).

The Sweet-Parker reconnection ansatz is undoubtedly correct. It has been simulated numerically many times, and was confirmed experimentally in the Magnetic Reconnection Experiment (MRX) operated by Princeton Plasma Physics Laboratory (PPPL) (Ji, et al. 1998). The problem is that Sweet-Parker reconnection takes place far too slowly to account for many reconnection processes that are thought to take place in the solar system. For instance, in solar flares $S\sim 10^{8}$, $V_A\sim 100\,{\rm km}\,{\rm s}^{-1}$, and $L\sim 10^{4}\,{\rm km}$ (Priest 1984). According to the Sweet-Parker model, magnetic energy is released to the plasma via reconnection on a typical timescale of a few tens of days. In reality, the energy is released in a few minutes to an hour (Priest 1984). Clearly, we can only hope to account for solar flares using a reconnection mechanism that operates far more rapidly than the Sweet-Parker mechanism.