Plasmoid Instability

One mechanism for obtaining fast magnetic reconnection (i.e., faster than Sweet-Parker reconnection) is via the plasmoid instability (Loureiro, et al. 2007; Bhattacharjee, et al. 2009). This instability causes Sweet-Parker current sheets to break up into chains of secondary magnetic islands (plasmoids).

Consider the linear stability of a current sheet of the form (9.17), whose thickness in the $x$-direction is $a$. It is helpful to define the Alfvén time, $\tau_A=a/(B_0^{\,2}/\mu_0\,\rho)^{1/2}$, as well as the modified Lundquist number, ${\cal S}= \tau_R/\tau_A$. Note that ${\cal S}$ has no dependence on the wavelength, $2\pi/k$, of the instability in the $y$-direction. Let us assume that $k\,a\ll 1$. It follows from Equation (9.39) that ${\mit\Delta}'\simeq 2/(k\,a)$. According to the analysis of Section 9.5, if the instability is in the constant-$\psi$ regime then

$$\gamma\,\tau_A = \left[\frac{\Gamma(1/4)}{\pi\,\Gamma(3/4)}\right]^{4/5}(k\,a)^{-2/5}\,{\cal S}^{-3/5},$$ (9.155)

$$\frac{\delta}{a} \sim (k\,a)^{-3/5}\,{\cal S}^{-2/5},$$ (9.156)

where $\delta$ is the linear layer thickness (Bhattacharjee, et al. 2009). The previous two equations are valid provided that $k\,a\,{\cal S}^{1/4}\ll 1$. On the other hand, according to the analysis of Section 9.6, if the instability is in the non-constant-$\psi$ regime then

$$\gamma\,\tau_A = (k\,a)^{2/3}\,{\cal S}^{-1/3},$$ (9.157)

$$\frac{\delta}{a} \sim (k\,a)^{-1/3}\,{\cal S}^{-1/3}.$$ (9.158)

The previous two equations are valid provided that $k\,a\,{\cal S}^{1/4}\geq 1$. Equations (9.155)–(9.158) suggest that the growth-rate of the instability attains a maximum value, $\gamma\,\tau_A\sim {\cal S}^{-1/2}$, when $k\,a\sim {\cal S}^{-1/4}$ and $\delta/a\sim {\cal S}^{-1/4}$. Note that the maximum growth-rate occurs at the boundary between the constant-$\psi$ and non-constant-$\psi$ regimes.

Figure: 9.11 Dispersion relation for the plasmoid instability. The horizontal and vertical dotted lines correspond to $\tilde{\gamma}=0.6235$ and $\tilde{k}=1.359$, respectively.

We can determine the maximum growth-rate exactly making use of the general linear dispersion relation (9.81). Let $\gamma\,\tau_A =\tilde{\gamma} \, {\cal S}^{-1/2}$ and $k\,a = \tilde{k}\,{\cal S}^{-1/4}$. For the case in hand, the dispersion relation yields

$$\tilde{k} = \left[-\frac{16}{\pi}\,Q^{-5/4}\,\frac{\Gamma(Q^{3/2}/4+5/4)}{\Gamma(Q^{3/2}/4-1/4)}\right]^{3/4},$$ (9.159)

where $\tilde{\gamma} =\tilde{k}^{2/3}\,Q$. Figure 9.11 shows the variation of $\tilde{\gamma}$ with $\tilde{k}$ obtained from the previous equation. It can be seen that $\tilde{\gamma}$ attains a maximum value of $0.6235$ when $\tilde{k}=1.359$. Thus, if $k$ is unconstrained then the fastest growing instability of a current sheet is such that $\gamma\,\tau_A = 0.6235\,{\cal S}^{-1/2}$ and $k\,a=1.359\,{\cal S}^{-1/4}$.

Let us now apply the previous analysis to a Sweet-Parker current sheet. According to Equations (9.84), (9.146), and (9.150), the thickness of a Sweet-Parker current sheet is

$$\frac{\delta_{\rm SP}}{a}\sim \left(\frac{1}{\epsilon\,\hat{W}\,{\cal S}}\right)^{1/2},$$ (9.160)

where $\epsilon = a/L$, and $L$ is the periodicity length of the primary magnetic island chain in the $y$-direction. (Thus, the wavenumber of the island chain is $k=2\pi/L$.) Let us assume that ${\mit\Delta}'\,\hat{W}\sim 1$, as is generally the case in non-constant-$\psi$ island chains. Given that ${\mit\Delta}'\sim 1/k$, it follows that $\hat{W}\sim \epsilon$. Hence,

$$\hat{\delta}_{\rm SP} \sim \epsilon^{-1}\,{\cal S}^{-1/2},$$ (9.161)

where $\hat{\delta}_{\rm SP} =\delta_{\rm SP}/a$. We can use the previous analysis to find the fastest growing instability of the Sweet-Parker current sheet by making the substitutions $a\rightarrow \hat{\delta}_{\rm SP}\,a$, $\tau_A\rightarrow \hat{\delta}_{\rm SP}\,a$, and ${\cal S}\rightarrow\hat{\delta}_{\rm SP}\,{\cal S}$. Thus, the fastest growing instability has a growth-rate

$$\gamma_p\,\tau_A \sim \hat{\delta}_{\rm SP}^{-3/2}\,{\cal S}^{-1/2}\sim \epsilon^{3/2}\,{\cal S}^{1/4},$$ (9.162)

and a wavenumber

$$k_p\,a \sim \hat{\delta}_{SP}^{-5/4}\,{\cal S}^{-1/4}\sim \epsilon^{5/4}\,{\cal S}^{3/8}.$$ (9.163)

It follows that the Sweet-Parker current sheet breaks up into a chain of secondary magnetic islands, or plasmoids, whose wavelength is $2\pi/k_p$. Thus, the number of plasmoids in the island chain is

$$N_p \sim L\,k_p \sim \epsilon^{1/4}\,{\cal S}^{3/8}.$$ (9.164)

The plasmoids are accelerated along the length of the current sheet and eventually expelled from its two ends at the Alfvén velocity. Recall, from Equation (9.154), that the linear growth-rate of the primary instability matches the nonlinear growth-rate at the boundary between the linear and nonlinear regimes. It is, therefore, reasonable to assume that the linear growth-rate of the secondary instability is similar in magnitude to its initial nonlinear growth-rate. This suggests that the plasmoid instability is able to reconnect magnetic flux at a much faster rate that the Sweet-Parker mechanism. In fact, the timescale for the plasmoid instablity to achieve full reconnection is estimated as

$$\tau_p \sim\frac{1}{\gamma_p} \sim \epsilon^{-3/2}\,\tau_A^{5/4}\,\tau_R^{-1/4}.$$ (9.165)

This is a significantly shorter timescale that that associated with Sweet-Parker reconnection,

$$\tau_{\rm SP} \sim \epsilon^{-1}\,\tau_A^{1/2}\,\tau_R^{1/2},$$ (9.166)

especially in a high Lundquist number plasma in which $\tau_R\gg\tau_A$. Note that the normalized thickness of the secondary Sweet-Parker current sheets that connect the plasmoids is

$$\hat{\delta}_p \sim \hat{\delta}_{\rm SP}^{3/4}\,{\cal S}^{-1/4}\sim \epsilon^{-3/4}\,{\cal S}^{-5/8}.$$ (9.167)

The previous analysis gives the impression that all Sweet-Parker current sheets are unstable to the plasmoid instability. In fact, this is not the case. In order for the analysis to remain valid there needs to be a reasonable separation between the thickness of the primary Sweet-Parker current sheet, $\delta_{\rm SP}$, and that of the secondary Sweet-Parker current sheets, $\delta_p$. Say,

$$\delta_p < \frac{1}{3}\,\delta_{\rm SP}$$ (9.168)

(Loureiro, et al. 2007). The previous relation leads to the plasmoid instablity criterion

$${\cal S} > (3\,\epsilon^{1/4})^8 \sim 10^4\,\epsilon^2.$$ (9.169)

In other words, the plasmoid instability only occurs when the Lundquist number exceeds a critical value that is of order $10^4$ (assuming that $\epsilon\sim 1$). This instability criterion has been verified numerically (Bhattacharjee, et al. 2009).