Plasmoid Instability in Forming Current Sheets
The plasmoid instability has revolutionized our understanding of magnetic reconnection in astrophysical environments. By preventing the formation of highly elongated reconnection layers, it is crucial in enabling the rapid energy conversion rates that are characteristic of many astrophysical phenomena. Most of the previous studies have focused on Sweet-Parker current sheets, which, however, are unattainable in typical astrophysical systems. Here, we derive a general set of scaling laws for the plasmoid instability in resistive and visco-resistive current sheets that evolve over time. Our method relies on a principle of least time that enables us to determine the properties of the reconnecting current sheet (aspect ratio and elapsed time) and the plasmoid instability (growth rate, wavenumber, inner layer width) at the end of the linear phase. After this phase the reconnecting current sheet is disrupted and fast reconnection can occur. The scaling laws of the plasmoid instability are not simple power laws, and depend on the Lundquist number (), the magnetic Prandtl number (), the noise of the system (), the characteristic rate of current sheet evolution (), as well as the thinning process. We also demonstrate that previous scalings are inapplicable to the vast majority of the astrophysical systems. We explore the implications of the new scaling relations in astrophysical systems such as the solar corona and the interstellar medium. In both these systems, we show that our scaling laws yield values for the growth rate, wavenumber, and aspect ratio that are much smaller than the Sweet-Parker based scalings.
It is now generally acknowledged that magnetic reconnection powers some of the most important and spectacular astrophysical phenomena in the Universe such as coronal mass ejections, stellar flares, non-thermal signatures of pulsar wind nebulae, and gamma-ray flares in blazar jets (Tajima & Shibata, 1997; Kulsrud, 2005; Zweibel & Yamada, 2009; Benz & Güdel, 2010; Ji & Daughton, 2011; Kumar & Zhang, 2015; Kagan et al., 2015). Although the importance of magnetic reconnection has been recognized since the 1950s, it has recently witnessed an upsurge in popularity due to the realization of the importance of the plasmoid instability in facilitating fast reconnection (and energy release).
The plasmoid instability can be understood in terms of a tearing instability occurring in a reconnecting current sheet (see Fig. 1). Numerical simulations providing clear indications that thin reconnecting current sheets may be unstable to the formation of plasmoids date back at least to the 1980s (Biskamp, 1982; Steinolfson & van Hoven, 1984; Matthaeus & Lamkin, 1985; Biskamp, 1986; Lee & Fu, 1986), but it is only in the last decade that their role in speeding up the reconnection process has been widely appreciated. Indeed, very narrow reconnection layers would form in the absence of the plasmoid instability, which, in turn, would have the effect of throttling the reconnection rate. However, reconnecting current sheets exceeding a certain aspect ratio cannot form because they become unstable to the formation of plasmoids, which break the reconnection layer into shorter elements, consequently leading to a significant increase in the reconnection rate (Daughton et al., 2006, 2009). Hence, the predictions of the classical Sweet-Parker reconnection model (Sweet, 1958; Parker, 1957) break down for sufficiently large systems such as those typically encountered in astrophysical environments - in these cases, it was shown that the reconnection rate becomes nearly independent of the magnetic diffusivity (Bhattacharjee et al., 2009; Huang & Bhattacharjee, 2010; Uzdensky et al., 2010; Loureiro et al., 2012; Huang & Bhattacharjee, 2013; Takamoto, 2013; Comisso et al., 2015a; Ebrahimi & Raman, 2015; Comisso & Grasso, 2016).
The ability of plasmoid-mediated reconnection to enable fast energy release has been exploited in explaining multiple phenomena in a wide range of astrophysical settings with considerable success. They include solar flares (Shibata & Tanuma, 2001; Bárta et al., 2011b, a; Li et al., 2015; Shibata & Takasao, 2016; Janvier, 2017), coronal mass ejections (Milligan et al., 2010; Karpen et al., 2012; Ni et al., 2012; Mei et al., 2012; Lin et al., 2015), chromospheric jets (Shibata et al., 2007; Ni et al., 2015, 2017), blazar emissions (Giannios, 2013; Sironi et al., 2015; Petropoulou et al., 2016; Beloborodov, 2017), gamma-ray bursts (Giannios, 2010; McKinney & Uzdensky, 2012; Kumar & Zhang, 2015) and non-thermal signatures of pulsar wind nebulae (Sironi & Spitkovsky, 2014; Guo et al., 2015; Werner et al., 2016; Sironi et al., 2016; Guo et al., 2016). Plasmoid formation can also produce self-generated turbulent reconnection (Daughton et al., 2011; Oishi et al., 2015; Huang & Bhattacharjee, 2016; Wang et al., 2016; Kowal et al., 2016), implying that large scale current sheets are likely to become turbulent during the advanced stages of the reconnection process (del Valle et al., 2016). Given the importance of nonlinear plasmoids in the reconnection process, several studies have also been devoted to the understanding of their statistical properties (Fermo et al., 2010; Uzdensky et al., 2010; Huang & Bhattacharjee, 2012; Guo et al., 2013; Shen et al., 2013; Janvier et al., 2014; Sironi et al., 2016; Lynch et al., 2016; Jara-Almonte et al., 2016; Lingam et al., 2017), which may be crucial to understand the occurrence of large abrupt events in solar, stellar and other massive objects flares (Shibata & Magara, 2011).
Although the impact of plasmoids in reconnection has been thoroughly documented, there are many fundamental issues that still remain unresolved. Several of them have to do with the linear phase of the plasmoid instability, for which a comprehensive dynamical picture is still missing. The linear phase of the instability is of fundamental importance because it allows us to understand in which conditions, and at what time, fast reconnection (which occurs when the plasmoids enter the nonlinear phase) is triggered. It is the goal of this paper to advance our theoretical understanding of the plasmoid instability in astrophysically relevant plasmas, but a historical background is first necessary to place our work in context with previous theoretical studies.
It is rather intriguing that the first derivation of the growth rate and the wavenumber for a special case of the plasmoid instability was presented in an exercise of a textbook. Indeed, this textbook by Tajima & Shibata (1997) showed that if one assumes that the reconnecting current sheet has an inverse-aspect-ratio corresponding to the Sweet-Parker one, , then the growth rate and the wavenumber of the plasmoid instability scale as and , respectively, where is the Lundquist number and is the Alfvénic timescale, being the magnetic diffusivity and the Alfvén speed calculated upstream of the current sheet. Therefore, the growth rate and the wavenumber (proportional to the number of plasmoids) increase monotonically with the Lundquist number of the system.
Surprisingly, these scaling relations remained overlooked until they were independently rederived by Loureiro et al. (2007) a decade later. They were also generalized further through the inclusion of three-dimensional (Baalrud et al., 2012), Hall (Baalrud et al., 2011) and plasma viscosity (Loureiro et al., 2013; Comisso & Grasso, 2016) effects. However, it soon became apparent that the results obtained by assuming a Sweet-Parker current sheet were problematic because of the growth rate being proportional to raised to a positive exponent, which implies that the growth rate approaches infinity in the limit . For sufficiently high -values, such as those typically encountered in astrophysical environments, the predicted growth rate would be so fast that the plasmoids would reach the nonlinear phase and disrupt the current sheet before the Sweet-Parker aspect ratio can be attained. This, of course, implies that a Sweet-Parker current sheet may not be realizable in the first place.
In order to circumvent this problem, Pucci & Velli (2014) conjectured that reconnecting current sheets break up when the condition is met 111The same condition was used to propose visco-resistive (Tenerani et al., 2015a) and collisionless (Del Sarto et al., 2016; Pucci et al., 2017) scalings. (more precisely ). This led them to an alternative set of scalings, including the result that the final inverse-aspect-ratio in the resistive regime depends only on the Lundquist number and corresponds to . A similar criterion was adopted in Uzdensky & Loureiro (2016), who posited that the linear stage of the plasmoid instability essentially ends when , with being the characteristic time scale of current sheet evolution. However, each of these assumptions are open to question. If one ‘terminates’ the linear dynamics at this stage, the growth rate is only comparable to the timescale of the current sheet thinning, implying that one cannot use the static dispersion relations of the tearing instability to carry out the calculations. Second, at this stage, the effects of the reconnection layer outflow are also non-negligible, since the modes are subject to stretching.
These difficulties are mostly rendered void when one observes that typically at the end of the linear phase. On top of that, it is also important to note that the most interesting dynamics occurs when . This has been shown in a recent Letter published by us (Comisso et al., 2016), where it was demonstrated that the plasmoid instability exhibits a quiescence period followed by a rapid growth. Furthermore, the scaling relations of the plasmoid instability were shown to be no longer simple power laws, as they included non-negligible logarithmic contributions and also depended upon the noise of the system, the characteristic rate of current sheet evolution, and even the nature of the thinning process. This has direct implications for the onset of fast magnetic reconnection, because the correct identification of the scaling laws of the plasmoid instability is necessary for understanding when and how plasmoids becomes nonlinear and disrupt the reconnecting current sheet.
In this work, we extend the analysis presented in the aforementioned Letter by formulating a detailed treatment of both the inviscid and viscous regimes of the plasmoid instability. A proper treatment of the latter is very important since viscosity (or equivalently, the magnetic Prandtl number, defined below) plays a major role in several astrophysical systems like accretion discs around neutron stars and black holes (Balbus & Henri, 2008), warm interstellar medium (Brandenburg & Subramanian, 2005), protogalactic plasmas (Kulsrud et al., 1997) and intergalactic medium (Subramanian et al., 2006). Our work accords four major advantages over prior studies: (i) the scaling laws for the plasmoid instability in general time-evolving current sheets are derived both in the resistive and visco-resistive regimes, (ii) a clear demarcation of the limited domain in which the previous scalings are applicable, (iii) the presentation of accurate results in astrophysically relevant regimes with very high -values, and (iv) the exploration of the astrophysical implications of the plasmoid instability in the stellar and interstellar medium contexts.
The outline of the paper is as follows. In Sec. 2, the least time principle, which is used to compute the properties of the dominant mode at the end of the linear phase, is introduced. This is followed by a derivation of the resistive and visco-resistive scaling laws for the plasmoid instability in Secs. 3 and 4, respectively. We discuss the astrophysical relevance of the derived scaling relations by choosing two systems (the solar corona and the interstellar medium) in Sec. 5. Finally, we summarize our results in Sec. 6.
2 Least time principle for plasmoids
In this Section, we provide a general framework to evaluate the properties of the plasmoid instability in general current sheets that can evolve over time. In such general current sheets, tearing modes (Biskamp, 2000; Goedbloed et al., 2010; Fitzpatrick, 2014) do not begin to grow at the same time, i.e. they are rendered unstable at different times. Moreover, their growth rate does not depend solely on the wavenumber , but also on the time that has elapsed since the current sheet evolution commenced at some initial aspect ratio (see example in Fig. 2).
The amplitude of the tearing modes evolves as per
where indicates the time-dependent growth rate, is the initial time and represents the perturbation from which the modes can start to grow. If were constant, the amplitude evolution would be identical to that obtained from conventional linear eigenmode theory. Since itself depends on time, it is more instructive to regard (1) as the WKB solution (Bender & Orszag, 1978) to the linearized equations governing the tearing mode process. Notice that in the linear phase, the amplitudes of the modes are ‘small’, and therefore they do not affect the current sheet evolution and there is no mode-mode coupling. Note also that we neglect mode stretching due to the reconnection outflow, which results to be a good approximation for the very large plasmas of interest in this work. We point out, however, that mode stretching is important to correctly evaluate the critical Lundquist number above which the plasmoid instability is manifested (see Huang et al., 2017).
The linear phase is terminated when the plasmoid half-width is on the same order as the inner layer width . The former is given by (see, for example, Fitzpatrick, 2014)
where , defined in (1), must be understood as being evaluated at the resonant surface. Here, represents the reconnecting magnetic field evaluated upstream of the current sheet, while is the half-width of the current sheet. The latter, namely , is not the same for all physics models, and hence we shall leave it unspecified at this stage.
As described above, we identify the end of the linear phase with the condition . Of course, this condition is not a sharp cutoff for the end of the linear phase. In reality, there could be a factor that needs to be included prior to the equality sign. However, for the purposes of simplicity, we shall terminate the linear phase when these two quantities are exactly equal to one another. Finally, a caveat regarding must be introduced - it represents the plasmoid half-width only when its associated mode is much more dominant than the rest. This assumption can be slightly relaxed to cases where the perturbation amplitude is sufficiently localized in the spectrum. It turns out that this condition is typically met at the end of the linear stage of the plasmoid instability. A more precise evaluation of the plasmoid width can still be obtained by considering the contribution of a proper range of the fluctuation spectrum (Huang et al., 2017).
Although we have now specified the end of the linear phase, we have not still identified the tearing mode that emerges dominant at the end. At this stage, we introduce the primary physical principle behind the paper. We follow the approach espoused in Comisso et al. (2016), namely, the mode that emerges “first” at the end of the linear phase is the one that has taken the least time to traverse it. This “principle of least time for the plasmoid instability” shares some apparent similarities with the renowned Fermat’s principle of least time, but there is also one essential difference - the latter relies upon a variational principle (Born & Wolf, 1980), whereas in the former the extremum of a function (the time) is computed.
Some modes may become unstable from an early stage and continue growing at a steady (relatively slow) pace. Others may remain stable for a long time, therefore remaining quiescent, until they become unstable at a later stage and are subject to explosive growth (see example in Fig. 3). Thus, amongst this wide range of possibilities, the above principle enables us to select the mode that exits the linear stage first. In mathematical terms, these conditions are expressible as follows. Firstly, we have
where the symbol ‘’ denotes the end of the linear phase. The above expression implies that the time is solely a function of the wavenumber . Then, the principle of least time amounts to stating that
Hence, the conditions (3) and (4) permit to compute the mode that takes the least value of . It can also be shown a posteriori that, in the neighborhood of , is localized and has a stronger dependence than on . Therefore, the mode that completes the linear phase in the least time can be seen as the dominant one that enters the nonlinear phase (see example in Fig. 4).
We can explicitly rewrite Eq. (3) as
where is the label for the initial perturbation amplitude and the functions and are defined such that and . We can also rewrite Eq. (4) as
if . Here we have defined
as in our previous Letter (Comisso et al., 2016).
Hitherto, our discussion has been completely general as the above relations (5) and (6) describing the principle of least time are equally applicable for a wide range of plasma models that can include resistive, viscous and collisionless contributions. In this paper, we shall focus primarily on the former duo, i.e. the resistive and visco-resistive regimes.
3 Resistive Regime
In what follows, we move to dimensionless quantities. We adopt a normalization convention such that all the lengths are normalized to the current sheet half-length , the time to the Alfvén time , and the magnetic field to the upstream field . Thus, the other quantities are normalized as
where we have used carets for denoting dimensionless quantities. Note that the normalized magnetic diffusivity corresponds to the inverse of the Lundquist number, i.e. , while the normalized kinematic viscosity corresponds to the inverse of the kinetic Reynolds number when the Alfvén velocity is the typical velocity scale of the system. Therefore, the ratio defines the magnetic Prandtl number. 222We specify that the magnetic Prandtl number that appears in the formulas of this paper is defined using the cross-field collisional value of .
In this Section we consider the plasmoid instability in the resistive regime, which is characterized by , while the next section is devoted to the visco-resistive regime, in which .
Although the framework provided in Sec. 2 is fully general, here we are interested in the case where the current sheet half-length and the reconnecting magnetic field remain approximately constant, while the current sheet width decreases in time. This is indeed a classic case of current sheet formation (see also Huang et al. 2017). The function that takes into account the current sheet thinning must obey and
When the modes grow slower than the evolution of the current sheet, i.e. , the change in is dominated by the change in and the growth rate is negligible in this respect. On the other hand, when the modes grow faster than the evolution of the current sheet, i.e. , the change in is mainly due to the growth rate of the perturbed magnetic flux. In this case, the tearing modes growth rate can be computed using the instantaneous value of in the standard tearing mode dispersion relations. Depending on the value of the tearing stability parameter (Furth et al., 1963), two simple algebraic relations can be considered. For (the small- regime), tearing modes grow as per the relation (Furth et al., 1963)
In our analysis, we are interested in the complete -domain. Therefore, we seek an expression for that (i) is a reasonable approximation of the exact dispersion relation (Coppi et al., 1976; Ara et al., 1978), (ii) reduces to (10) and (11) in the proper limits, and (iii) is simple enough to be analytically tractable. For this purpose we adopt the half-harmonic mean of this two relations, namely
which fulfills all of the criteria described above. If a different approximation for is adopted, such as the simpler one employed in several previous works (Tajima & Shibata, 1997; Bhattacharjee et al., 2009; Loureiro et al., 2013; Huang & Bhattacharjee, 2013; Pucci & Velli, 2014), or the more complex one used by Huang et al. (2017), the same scaling relations as the ones derived below are obtained, albeit with slightly different numerical factors.
At this point we only need to specify the inner resistive layer width, which corresponds to (Fitzpatrick, 2014)
where . In our subsequent discussion, we assume that the natural noise of the system has a general power law form, namely , but other cases can be treated on an equal footing by considering different perturbation spectra. We also assume that the current sheet has the common Harris-type structure (Harris, 1962), for which . This expression for can be simplified by considering only the regime , since the very slow-growing part of the mode evolution does not affect the results of the linear phase. Then, from Eq. (14) we get
It can be shown a posteriori that the two terms on the right-hand-side must approximately balance each other for . Hence, the emergent mode satisfies the relation
where is a coefficient. This implies that
where and are also coefficients. The above relations indicate that the dominant mode at the end of the linear phase has the same scaling properties of the fastest growing mode (Furth et al., 1963), the latter of which satisfies the equation .
For moderately high values of the Lundquist number , the reconnecting current sheet might be capable of attaining the Sweet-Parker inverse-aspect-ratio . In this case, it is straightforward to see that
It is therefore not surprising to discover that these relations match the ones obtained in previous studies of the plasmoid instability (Tajima & Shibata, 1997; Loureiro et al., 2007; Bhattacharjee et al., 2009; Baalrud et al., 2012; Huang & Bhattacharjee, 2013; Comisso & Grasso, 2016) that were undertaken assuming a fixed Sweet-Parker current sheet.
On the other hand, for very high Lundquist numbers, which are widely prevalent in most astrophysical plasmas, the plasmoids complete their linear evolution well before the Sweet-Parker aspect ratio is reached. Thus, we need to calculate for a more general case. This can be done by substituting the relations for the dominant mode at the end of the linear phase into Eq. (5), which yields the following equation for the inverse-aspect-ratio:
This equation gives us the final inverse-aspect-ratio for a general current sheet evolution . It is evident that , and thus the scaling relations of , , and , cannot be universal, because they depend on the specific form of the function .
Since we must specify a specific form of , in what follows, we first consider what is probably the most typical case of current sheet thinning, namely the exponential thinning. This is indeed known to be standard case for instability-driven current sheets. Then, we generalize the results of the exponential thinning to include also algebraic cases. Other, less common, possibilities could also be investigated, since the developed framework is general.
3.1 Exponentially shrinking current sheet
It can be shown from first principles that the exponential thinning of a reconnecting current sheet evolves according to the expression (Kulsrud, 2005)
where in the resistive regime (Sweet, 1958; Parker, 1957). This expression slightly differs from the one we adopted in our previous work (Comisso et al., 2016), but it shares the same asymptotic behaviors for small and large , therefore leading to the same asymptotic relations for the plasmoid instability. Other cases for , such as the ones imposed in the numerical simulations by Tenerani et al. (2015b), where , are not considered here, because they are not supported by physical evidence (Kulsrud, 2005; Huang et al., 2017). In this respect, it is worth noting that the plasmoid half-width (2) starts to grow only when (for it is straightforward to check that decreases because of the rapid decrease of ). We will see later that this condition occurs when , which is smaller than for of order unity, which is indeed the case for an ideal exponentially thinning current sheet (Kulsrud, 2005; Huang et al., 2017).
Using Eq. (21) we can compute the transitional time that separates the two asymptotic behaviors for small and large . For , with
we have . On the other hand, for we have . While for one recovers the relations (19), the case , which is more relevant for astrophysical environments because it occurs for larger -values, necessitates further analysis. In this case we have to solve Eq. (20). Using , we obtain
The coefficient turns out to be for of order unity. Therefore, we can neglect the factor in Eq. (23). This equation for the inverse-aspect-ratio can be solved exactly in terms of the Lambert function (Corless et al., 1996), but here we prefer to consider an asymptotic solution that yields more transparent results. As was done in Comisso et al. (2016), we solve Eq. (23) by iteration obtaining
and the subdominant term proportional to has been neglected.
Given the final inverse-aspect-ratio, we can easily determine the growth rate, wavenumber and inner layer width at the end of the linear phase:
These relations exhibit a non-trivial dependence with respect to the Lundquist number , the noise level (through both and ), and the timescale of the driving process . Note that these scaling relations of the plasmoid instability are not pure power laws, as they also include non-negligible logarithmic factors. This has important implications for very large- plasmas like those typically encountered in astrophysical environments (Ji & Daughton, 2011), since the scaling properties of the plasmoid instability change considerably with respect to those obtained for not-so-large plasmas.
To better evaluate the implications of the new scaling relations, let us focus on the case in which the natural noise amplitude is approximately the same for all wavelengths. In this case we can set , and, recalling that , Eqs. (26)-(30) reduce to
Note that these expressions are identical to those obtained in Comisso et al. (2016) once that the quantity is not explicitly set to unity.
From Eq. (31), together with Eq. (21), we can see that the final inverse-aspect-ratio turns out to be bounded between . Eq. (31) also indicates that decreases for smaller perturbation amplitudes . The final inverse-aspect-ratio as a function of for two different values of is plotted in Fig. 5. An inspection of this figure reveals that the Sweet-Parker aspect ratio can be attained only for moderately high -values. The domain of existence of the Sweet-Parker aspect ratio may be slightly extended in lower noise systems, but, for most of the astrophysically relevant regimes, the final width of the reconnecting current sheet remains thicker as predicted by Eq. (31).
The dependence of the growth rate and the wavenumber as a function of the Lundquist number change significantly upon considering large systems. This is clearly shown in Figs. 6 and 7, where the black dashed lines represent the earlier scalings, which are clearly not applicable to large- plasmas, while the solid curves represent the results that have been obtained by means of this new theoretical approach. The behavior of is non-monotonic in , while displays a monotonic behavior, but with much lower values with respect to the Sweet-Parker-based solution for large values of . While counterintuitive at first glance, the decrease of the final growth rate for very large can be understood by noting that the inner layer width decreases for increasing , therefore, a given noise amplitude leads to perturbation amplitudes closer to the condition for the end of the linear phase if is larger. This, in turn, reduces the time available for the acceleration of the perturbation growth.
Our approach also enables a quantification of the effects of noise: lower values of the noise can increase the final instantaneous growth rate and the number of plasmoids (which is proportional to ), as can be seen from the orange curves in Figs. 6 and 7. Finally, note that for large- astrophysical environments, Eq. (34) (not plotted here) indicates that the inner resistive layer width at the end of the linear phase is thicker than what would be predicted using the Sweet-Parker-based solution (19).
An important observable that can be duly obtained from Eq. (26) is the time that has elapsed since the current sheet evolution commenced at the initial inverse-aspect-ratio . This timescale corresponds to
Fig. 8 shows that the elepsed time computed by means of the principle of least time has a non-monotonic behavior, confirmed also by recent numerical simulations (Huang et al., 2017), and after reaching a minimum value at moderate -values, it increases as predicted by Eq. (35). Note that the time does not correspond to the time required for the plasmoids to grow. This is due to the fact that the final (dominant mode) wavenumber remains quiescent for a certain period of time before being subject to violent growth over a short timescale (as shown in the example in Fig. 3).
The actual time that it takes for the final wavenumber to undergo finite growth is , where is the inverse-aspect-ratio at the onset time, i.e. when . Using Eq. (12) and retaining the dominant terms, we find . Therefore, the intrinsic timescale of the plasmoid instability becomes
This timescale exhibits a very weak dependence on the Lundquist number and the natural noise of the system, meaning that the intrinsic timescale of the plasmoid instability is nearly universal for exponentially thinning current sheets.
Finally, we want to evaluate the value of the Lundquist number above which the scaling laws of the plasmoid instability change behavior as described by the previously obtained equations. We refer to this value as to the transitional Lundquist number. It can be obtained by equating the two asymptotic behaviors of , which yields the equation
The exact explicit solution of this equation is
where , and is the Lambert function, which is defined such that . This expression exhibits a complex dependence on the noise level ( and ), and the timescale of the driving process (). A simpler asymptotic approximation can be constructed when considering large arguments of the Lambert function. In this case (Corless et al., 1996)
Keeping only the first term of this expansion, we obtain
From this expression we can see that decreases if the timescale of the current sheet thinning becomes larger. Furthermore, an increase of occurs for lower values of and/or increasing values of . The accurate behavior of the transitional Lundquist number as a function of the system noise for a wide range of noise amplitudes is shown in Fig. 9. The transitional Lundquist number turns out to be fairly modest even for very low noise amplitudes, implying that the plasmoid instability in most of the astrophysical systems should follow the newly obtained scaling laws.
3.2 Generalized current sheet shrinking
It is possible to generalize the results obtained for exponentially thinning current sheets in order to also describe current sheets whose thinning depends algebraically on time. While the former is the natural consequence of an instability-driven current sheet, the latter has been shown to occur in several cases of forced magnetic reconnection (Hahm & Kulsrud, 1985; Wang & Bhattacharjee, 1992; Fitzpatrick, 2003; Bhattacharjee, 2004; Birn et al., 2005; Hosseinpour & Vekstein, 2008; Comisso et al., 2015a, b). To encompass both exponential and algebraic behaviors, we consider a generalized current shrinking function of the form
where . This expression recovers the exponential thinning specified in Eq. (21) when taking the limit , while other cases can be obtained by considering different values of . For example, setting , we obtain a current sheet thinning that is inversely proportional in time. This is relevant for various forced reconnection models, most notably the Taylor model (Hahm & Kulsrud, 1985; Wang & Bhattacharjee, 1992; Fitzpatrick, 2003; Comisso et al., 2015a, b; Zhou et al., 2016; Beidler et al., 2017), which has applications in both laboratory and astrophysical plasmas.
For the adopted generalized current thinning function, the plasmoid half-width (2) starts to grow when . Furthermore, the transitional time that separates the two asymptotic solutions for the plasmoid instability is
As before, we are especially interested in the case , since it is astrophysically more relevant. However, to derive the analytical solution in this case, we consider only the small branch of the dispersion relation in consideration of the fact that , where is the instantaneous growth rate of the fastest growing mode. Thus, we approximate with in Eq. (20), using again and . Therefore we obtain
Solving this equation along the same lines as Eq. (26), we obtain
From Eq. (45), we obtain the following generalized scaling relations for the plasmoid instability:
For , we recover the scaling relations (26)-(30), while different choices of give us the scaling relations relevant for different algebraic thinning possibilities. These expressions indicate that faster current sheet shrinking rates (larger and/or smaller ) lead to larger aspect ratio (), growth rate and wavenumber. On the other hand, the inner resistive layer width at the end of the linear phase decreases for faster current sheet formation. It is also interesting to observe that for (i.e. ), the number of plasmoids increases with in the astrophysically relevant regimes, but the opposite trend is possibly manifested for . In other words, for the latter case, the number of plasmoids can actually decrease in this regime as increases. For , where the thinning is inversely dependent on the time, the scaling of the number of plasmoids with is weak (logarithmic).
As it should be expected, the elapsed time from the initial aspect ratio can change significantly for different current sheet formation rates. Indeed, from Eqs. (45) and (41), with , the elapsed time results to be
Lower values of lead to much higher values of the elapsed time , implying that the final dominant wavenumber remains quiescent for a much longer period of time when the current sheet evolution is slower.
The transitional Lundquist number for this class of generalized thinning current sheets can be computed by equating the two asymptotic branches for in a manner analogous to that of exponential thinning. Thus, we are led to the equation
which can be inverted in a straightforward manner, by means of the Lambert function, to obtain . Similarly, it is also possible to compute the timescale for the plasmoid instability in this generalized scenario by following the procedure outlined for exponential thinning sheets.
We shall explore the implications of our preceding results for astrophysical plasmas in Section 5. Next, we consider the visco-resistive regime and carry out a similar analysis.
4 Visco-Resistive Regime
In this Section, we derive the corresponding scaling laws of the plasmoid instability in the presence of strong plasma viscosity, namely when . Plasma viscosity is indeed important in several astrophysical environments such as the (i) warm interstellar medium, (ii) protogalactic plasmas, (iii) intergalactic medium and (iv) accretion discs around neutron stars and black holes (Kulsrud & Anderson, 1992; Brandenburg & Subramanian, 2005; Balbus & Henri, 2008). In this case . Furthermore, two different relations for the growth rate in the small- and large- regimes must be considered. For , the growth rate of the tearing modes modified by strong plasma viscosity is (Bondeson & Sobel, 1984)
where . On the other hand, for the growth rate satisfies the relation (Porcelli, 1987)
where . An effective approximation for across the entire domain of can be constructed as before, by using Eq. (12).
it is possible to obtain the least time equation
which follows from a careful application of Eq. (7). By repeating the procedure delineated in the previous Section, we can find the counterpart of Eq. (3) that is valid in the visco-resistive regime. This corresponds to
It can be shown a posteriori that for , the two terms on the right-hand-side must approximately balance each other. Thus, we end up with
where , and are coefficients.
By assuming that the reconnecting current sheet has the time to reach the asymptotic value , one can find