Analytic scaling relations are derived for a phenomenological model of the plasmoid instability in an evolving current sheet, including the effects of reconnection outflow. Two scenarios are considered, where the plasmoid instability can be triggered either by an injected initial perturbation or by the natural noise of the system (here referred to as the system noise). The two scenarios lead to different scaling relations because the initial noise decays when the linear growth of the plasmoid instability is not sufficiently fast to overcome the advection loss caused by the reconnection outflow, whereas the system noise represents the lowest level of fluctuations in the system. The leading order approximation for the current sheet width at disruption takes the form of a power law multiplied by a logarithmic factor, and from that, the scaling relations for the wavenumber and the linear growth rate of the dominant mode are obtained. When the effects of the outflow are neglected, the scaling relations agree, up to the leading order approximation, with previously derived scaling relations based on a principle of least time. The analytical scaling relations are validated with numerical solutions of the model.
Thin current sheets, which appear to be ubiquitous in laboratory, space, and astrophysical plasmas, are known to be unstable to the plasmoid instability,1–3 which disrupts current sheets to form smaller structures such as plasmoids (or flux ropes) and secondary current sheets. Plasmoid-instability-mediated disruption of reconnecting current sheets plays a crucial role in triggering the transition from slow to fast magnetic reconnection3–9 and also modifies the energy cascade in magnetohydrodynamic (MHD) turbulence, making the energy power spectrum steeper than that predicted by traditional MHD turbulence theories.10–17
In early theoretical studies of the plasmoid instability, it was commonly assumed that the aspect ratio of the current sheet follows the Sweet-Parker scaling ,18,19 where S ≡ LVA/η is the Lundquist number. Here, L and a denote the half-length and the half-width of the current sheet, respectively, VA is the Alfvén speed, and η is the magnetic diffusivity. For the resistive magnetohydrodynamics (MHD) model, this assumption yields the scaling relations γ ∼ S1/4 for the linear growth rate and k ∼ S3/8 for the wavenumber of the fastest growing mode.2,3,20 Following the same assumption, scaling relations have also been derived for other models, including Hall MHD21 and viscoresistive MHD.22
However, a current sheet in nature typically forms dynamically, starting from a broader sheet that gradually thins down. The fact that the Sweet-Parker-based scaling γ ∼ S1/4 diverges in the asymptotic limit of S → ∞ indicates that for a high-S current sheet, the disruption may occur before the current sheet realizes the Sweet-Parker aspect ratio.23 The precise conditions for current sheet disruption have been a topic of active research in recent literature.9,23–29 By using a principle of least time, Comisso et al. showed that the current sheet width, the linear growth rate, and the dominant mode wavenumber at disruption do not follow power-law scaling relations.27,28 Huang et al. proposed a phenomenological model incorporating the effects of reconnection outflow.9 Numerical solutions of this model have been extensively tested with results from direct numerical simulations.
The objective of this work is to develop a methodology to obtain analytical scaling relations for the phenomenological model of Huang et al. for a special class of evolving current sheets, which is modeled by a Harris sheet thinning down exponentially in time, where the upstream magnetic field remains constant. This paper is organized as follows. Section II gives an overview of the model. Section III gives a formal solution of the model equation based on the method of characteristics. The solution is then applied in Sec. IV to obtain scaling relations for key features at the disruption time, such as the current sheet width, the linear growth rate, and the dominant mode wavenumber. Here, we consider two possible scenarios. In the first scenario, an initial noise is injected into the system to trigger the plasmoid instability. In the second scenario, the plasmoid instability evolves from the natural noise of the system, referred to as the “system noise” in this paper.30 The two scenarios lead to different scaling relations because the initial noise tends to decay during an early time when the linear growth of the plasmoid instability is not sufficiently fast to overcome the advection loss caused by the reconnection outflow; on the other hand, the system noise represents the lowest level of fluctuations pertaining to the system even when no external noise is injected. In Sec. V, we consider the case where the effects of the outflow are neglected. In this case, we show that the scaling relations agree with that of Comisso et al. up to the leading order approximation. We also validate the analytical scaling relations by comparing them with numerical solutions. We conclude in Sec. VI.
II. PHENOMENOLOGICAL MODEL
In this section, we give a brief outline of the phenomenological model. The reader is referred to Ref. 9 for the details of the derivation. Figure 1 shows a schematic diagram for the system we consider here. The following equation governs the linear phase of the plasmoid instability in an evolving current sheet:
Here, is the Fourier amplitude of the magnetic field fluctuation normal to the current sheet as a function of the wavenumber k and the time t, normalized to the characteristic magnetic field B0 and length L0 of the system, vo is the outflow speed, γ(k, t) is the linear growth rate of the tearing instability, and L is the half-length of the current sheet. Here, the linear growth rate γ explicitly depends on time because the current sheet can evolve in time. The differential operator in the left-hand-side incorporates the stretching effect on wavelength due to the reconnection outflow; the right-hand-side represents the effects of linear growth, advection loss due to the outflow, and evolution of the current sheet length. The domain of wavenumber k is limited to k ≥ π/L because the wavelength must be smaller than the current sheet length 2L.
The linear growth rate γ(k, t) depends on the profile of magnetic field Bx(z, t), which can be a function of time depending on the system under consideration. The model also allows the current sheet half-length L to change in time, as represented by the (1/2L)dL/dt term. The outflow speed v0 can also be time-dependent. To integrate the model equation, we need γ(k, t), vo(t), L(t), and an initial condition f = f0(k) as inputs. With these conditions provided, we can integrate equation (1) in time until the plasmoid instability enters the nonlinear regime, which occurs when the typical half-width w of plasmoids exceeds the inner layer half-width δ. At that moment, the fluctuating part of the current density is of the same order of the background current density J, implying that the current sheet has lost its integrity. Therefore, we take w = δ as the condition for current sheet disruption.
This condition for current sheet disruption depends critically on the half-width w of plasmoids. However, the precise definition of w is nuanced. The nuance comes from the fact that while w is well-defined when the magnetic fluctuation is a single-mode Fourier harmonic (as is often assumed in tearing mode analyses), it is not so when the fluctuation is a superposition of a continuum of Fourier modes. In this model problem, fortunately, a meaningful half-width w of plasmoids can be defined because, as we will see, the solution f typically becomes localized around a dominant wavenumber kd as time progresses. For a spectrum f(k) localized around k = kd, we define the plasmoid half-width w as
where the prime denotes d/dz evaluated at z = 0. In Eq. (2), the fluctuation amplitude must take into account the contribution from all the neighboring modes of the dominant mode. Precisely, a superposition of all the modes in the range k ∈ [kd/ξ, kdξ] gives a fluctuation amplitude
Here, the O(1) parameter ξ sets the range of superposition. As the Fourier spectrum f(k) is usually well-localized at the disruption time, the amplitude is insensitive to the choice of ξ, provided that ξ is not too close to unity (e.g., ξ = 1.5 was used in Ref. 9). Note that for a single-mode perturbation, definition (2) reduces to the usual expression for the magnetic island half-width.31
The plasmoid half-width w must be compared with the inner layer half-width δ to determine the condition for current sheet disruption. As the inner layer half-width δ depends on the wavenumber, we adopt the inner layer half-width at the dominant wavenumber kd for this comparison. For resistive tearing modes we consider in this study, the inner layer half-width for k = kd is given by
where η is the magnetic diffusivity. Here, the prime again denotes d/dz evaluated at z = 0, and VA is the Alfvén speed of the reconnecting component Bx. The criterion w = δ, with w and δ defined by Eqs. (2)–(4), then determines the condition for current sheet disruption. This criterion has been extensively tested and validated with numerical simulations.9
III. INTEGRAL SOLUTION OF THE MODEL EQUATION FOR A THINNING HARRIS SHEET
Until now, the model has been general regarding the time evolution of both the current sheet profile and the reconnection outflow. To fix ideas and make the model analytically tractable, we assume that the current sheet can be modeled by a Harris sheet and the time evolution of the width a obeys the form
In this form, assuming a∞ ≪ a0, the current sheet width a decreases exponentially at early times with , and a → a∞ as t → ∞. We further assume that the upstream magnetic field Bx and the current sheet half-length L remain fixed in time and that the outflow speed vo is the upstream Alfvén speed VA. Under these conditions, if the plasmoid instability does not disrupt the current sheet, eventually the current sheet will approach the Sweet-Parker current sheet, which is a steady-state solution of reconnection governed by resistive MHD. Therefore, we set the asymptotic half-width a∞ to be the Sweet-Parker width, i.e., , where S ≡ LVA/η is the Lundquist number. These assumptions are consistent with direct numerical simulations reported in Ref. 9. Heuristic derivations for Eq. (5) can be found in Refs. 32 and 9.
Under the assumption that the upstream Bx and the current sheet length L are fixed in time, it is natural to choose B0 = Bx and L0 = L as the characteristic magnetic field and length scale. Hereafter, we normalize lengths and times with respect to the current sheet half-length L and the Alfvén time τA ≡ L/VA, as follows: , and . The model equation in normalized variables is
where dL/dt = 0 is assumed.
We can solve Eq. (6) by the method of characteristics. The characteristic starting from at is given by
which represents the mode-stretching effect due to the reconnection outflow. Now the differential operator on the left-hand-side of Eq. (6) is simply the time derivative along characteristics and hence the solution can be obtained by integrating along them, yielding
To perform the integration, we need the linear growth rate as a function of the wavenumber and time. The linear growth rate of the tearing mode is governed by the tearing stability index ,33 which depends entirely on the solution of linearized ideal MHD force-free equation in the outer region away from the resonant surface (where k⋅B = 0). The tearing instability requires the condition being satisfied. While the complete dispersion relation for resistive tearing modes can be expressed in terms of , the expression is involved and requires solving a transcendental equation.34 Analytically tractable approximations can be obtained in two limits: The small- regime for short-wavelength modes and the large- regime for long-wavelength modes. In the small- regime (large k and ), the linear growth rate is given by33
where . In the large- regime (small k and ), the linear growth rate is approximately given by34
Here, the derivative is evaluated at the resonant surface. For a Harris sheet profile with , the tearing stability index is given by
From Eqs. (9)–(11), we obtain the normalized growth rates for a Harris sheet in the small- regime
and in the large- regime
for an arbitrary ζ. In this work, we take the value ζ = 3/2, which gives a nearly exact approximation of the true dispersion relation. Furthermore, we will neglect the factor in Eq. (12) for our analytical derivations. This approximation can be justified a posteriori by noting that the dominant mode wavenumber follows the same scaling as the fastest growing mode wavenumber [see Eq. (35) and the discussion thereafter] and that (see below); therefore, in the high-S regime, the condition is satisfied in the neighborhood (in the k-space) of the dominant mode, which is our primary interest. Note that, from the scaling of the disruption current sheet width, Eq. (47), the condition is satisfied. Putting these approximations together yields an approximate expression for the linear growth rate that is valid for , i.e.,
The fastest growing mode occurs at the transition between the small- and the large- regimes, i.e., at where . More precisely, the fastest growing wavenumber is given by (see, e.g., Ref. 35)
and the corresponding growth rate is
IV. SCALING RELATIONS FOR CURRENT SHEET DISRUPTION
In this section, we derive scaling relations for key features at current sheet disruption in the high-S regime, under the assumptions detailed in Sec. III. Specifically, the disruption time , the current sheet width , the dominant mode wavenumber , and the linear growth rate of the dominant mode are derived analytically. Here, we consider two cases. In the first case, an initial noise is injected into the system. The initial noise is assumed to be much larger than the system noise; therefore, the latter is ignored and set to zero in the analysis. In the second case, there is no initial noise, and the system noise is the seed from which the plasmoid instability develops.
A. Case I: Initial perturbation
Let the initial perturbation be at . To integrate the linear growth rate along a characteristic in Eq. (8), it is convenient to define a new variable
In the high-S regime we consider here, it can be shown a posteriori that current sheet disruption occurs when , because follows a weaker scaling with respect to S [see Eq. (47)] than , which is assumed to be the Sweet-Parker width . Hence, we will ignore a∞ in Eq. (5) and assume in the following analysis. The combined effect of mode stretching ( ) and current sheet thinning implies that the variable φ decreases exponentially in time as
where the time scale is defined as
The linear growth rate, Eq. (15), can be expressed in terms of φ instead of as
We can now integrate the growth rate along the characteristic with a change of variable from to φ, yielding
Here, 2F1 is the hypergeometric function36 and
To proceed from Eq. (22), we further make the approximation that in the high-S regime, the contribution from the lower boundary, , is negligible compared to the contribution from the upper boundary, , at the disruption time. This approximation can be justified a posteriori using the scaling relation of the disruption current sheet width in Eq. (47). Because decreases as S increases, it follows from the relation that in the high-S regime, the condition φ ≪ φ0 is satisfied at the disruption time, where the variable φ is typically an O(1) quantity in the neighborhood of the dominant mode [see the discussion in the paragraph below Eq. (34)]. The function is monotonically decreasing with respect to . Its overall behavior is well captured by two asymptotic expressions for small and large , which can be obtained using the asymptotic expansion of the hypergeometric function near x → ∞
and the Taylor expansion near x = 0
for the special case with b = c, Eq. (25) is not valid and must be replaced by
It follows that the function is approximately
when , and
when . For the special case that yields , Eq. (28) is replaced by
Because φ0 ≫ φ ∼ O(1), it follows from Eqs. (28)–(30) that .
Ignoring , we can simplify Eq. (22) as
The solution of the model equation can now be written as
with , and being substituted into the right-hand side.
The dominant wavenumber at the disruption time can be obtained by locating the maximum of f. Because the exponential factor in Eq. (34) is highly localized, whereas most random fluctuations have a broadband spectrum, we further assume that is slowly varying compared to the exponential factor. Under this assumption, the dominant wavenumber is approximately the one that maximizes the exponent in Eq. (34).37 Let φ = φd correspond to the maximum of . The value of φd depends on and, in general, has to be obtained numerically; typically, φd is an O(1) quantity. With the value of φd obtained, the dominant wavenumber and linear growth rate are given by
where the constants ck and cγ are defined as
Equation (35) shows that the dominant wavenumber and the fastest growing wavenumber follow the same scaling [cf. Eq. (16)]. However, as we will see in Table I, typically the constant ck is smaller than the coefficient in Eq. (16). Therefore, the dominant wavenumber is smaller than the fastest-growing wavenumber. This discrepancy between the dominant wavenumber and the fastest-growing wavenumber is due to the fact that the mode amplitude at a given wavenumber is determined by the entire history of the linear growth rate following the characteristic. Even though the dominant mode is not the fastest growing mode at the moment, it has higher linear growth rates at earlier times.9
|.||χ = 0 .||χ = 0 .||χ = 1 .||χ = 3 .|
|.||χ = 0 .||χ = 0 .||χ = 1 .||χ = 3 .|
Equation (35) gives the relation between and but we still need to determine the latter. This final step is accomplished by using the disruption condition . From Eqs. (2) and (4), the disruption condition is equivalent to
where the magnetic fluctuation is calculated by integrating the fluctuation spectrum f over a neighborhood of the dominant wavenumber as prescribed in Eq. (3). Because the fluctuation spectrum f is localized around the dominant wavenumber, we can first expand the exponent in Eq. (34) near and then approximate the integration by a Gaussian integral. Expanding g in the neighborhood of yields
Hence, in the neighborhood of the dominant mode,
and is the disruption time. Now we can evaluate the magnetic fluctuation by using Eq. (41) and extending the domain of integration to (−∞, ∞) (see Fig. 2 for an illustration of this approximation), yielding
Equation (44) is the fundamental equation to determine the disruption current sheet width for a general initial condition ; its validity only requires that is slowly varying such that the dominant wavenumber in the spectrum, given by Eq. (34), is determined by the peak in the exponent. However, Eq. (44) is difficult to solve for a general without resorting to numerical methods. To make further progress, we assume that the initial condition takes a power-law form . Then, Eq. (44) becomes
An a posteriori consistency check for the Gaussian integral approximation in Eq. (43) is in order. Note that g(φd) and are typically O(1) quantities; therefore, the validity of the Gaussian integral approximation requires the condition being satisfied. Using Eq. (47) in Eq. (32), the condition is
This condition is satisfied only if . Therefore, the noise amplitude ϵ must be sufficiently small for the approximation to be valid.
Because the condition is satisfied, we can use the asymptotic expansion of W–1(z) as z → 0−, i.e.,
to obtain the leading order approximation for
Here, we have ignored an O(1) factor within the logarithm, which can be attributed to the next order correction of the form .
B. Case II: System noise
The calculation in Sec. IV A assumes that an initial perturbation is applied at . If the current sheet width is sufficiently broad such that the linear growth rate , and then the perturbation amplitude at wavenumber decreases in time because of the advection loss caused by the reconnection outflow [Eq. (6)]. If the linear growth rate never rises above 1/2, then the perturbation amplitude will decrease monotonically in time and asymptotically approach zero as . However, since noise is present in any natural system, there will always be some fluctuations in the system. Noise can be introduced into the model by explicitly adding a source term or by setting a lower bound (as a function of ) to the fluctuation amplitude. Here, we adopt the second approach: Instead of an initial perturbation, we now use to describe the system noise, i.e., represents the lower bound of ). In this section, we consider the situation where the system noise provides the seed for the plasmoid instability.
Because must be greater than 1/2 for a mode to grow when we integrate along a characteristic in Eq. (8), the starting time is determined by the condition . For neighboring modes of the dominant mode at the disruption time, it can be shown a posteriori that those modes start to grow when they are in the small- regime.39 Using the relations and in the condition using the small- dispersion relation
hence, the wavenumber when the mode starts to grow is
Following the same calculation leading to Eq. (34), the solution is
The disruption condition (39) yields the equation to determine
If we assume a power-law system noise , Eq. (59) becomes
The solution for can be expressed in terms of the lower-branch Lambert function W−1 as
Here, we again ignore an O(1) factor within the logarithm, which can be attributed to the next order correction of the form log(log(…)).
A. Effects of the reconnection outflow
Our phenomenological model includes the effects of the reconnection outflow. It is instructive to investigate how the outflow affects the scaling relations. The effects of the outflow can be turned off by setting vo to zero in Eq. (1). While we have repeated the calculation for cases without the outflow, here we present an easier way that yields the same results. Neglecting the effects of the outflow can be formally achieved by assuming that the current sheet thins down almost instantaneous, i.e., the current sheet thinning time scale τ ≪ τA, such that the mode-stretching and advective loss due to the outflow have no time to take effect. Taking the limit is thus equivalent to neglecting the effects of the outflow. In this limit, the scaling relations obtained in Secs. IV A and IV B reduce to the same results; i.e., it makes no difference whether the plasmoid instability evolves from an initial noise or a system noise if the effects of the outflow are ignored.
In the limit, the two time scales and become identical to . The variables θ and Ξ become
The disruption current sheet width is given by
with the leading order approximation
B. Comparison with the scaling relations of Comisso et al.
Now we are in a position to compare the analytical scaling relations in this work with that obtained previously by Comisso et al.,27,28 because the latter also ignore the effects of outflow. However, as Comisso et al. used a different way to describe the fluctuations, we need a translation of notations before a comparison can be made.
Comisso et al. described the fluctuations by the island size as a function of and , where is the magnetic flux of the island. They propose a “principle of least time” that determines the dominant mode to disrupt the current sheet by the wavenumber that satisfies the condition with the shortest time. For an initial condition , the leading order approximation of the current sheet width at disruption is given by Eq. (26) of Ref. 28
The present work describes the fluctuations by a spectrum of the magnetic field. To compare Eq. (70) with Eq. (69), we need to find a correspondence between f and . This can be obtained by noting that the magnetic field fluctuation [from Eq. (3)] and [from Eq. (2) and the relation ]; hence, the correspondence is . For the assumed power-law initial perturbations and , we have the correspondence . Substituting α → χ + 1/2 into Eq. (69), we recover the scaling relation (70). Therefore, we conclude that when the effects of the outflow are ignored, the phenomenological model of this work yields the same scaling relations as the relations obtained using the principle of least time. However, the agreement is only up to the leading order approximation. It can be shown that the two approaches give different results if the next order corrections of the Lambert W function of the form log(log(…)) are included.
C. Comparison with numerical solutions of the model equation
We have employed several approximations when deriving the analytical scaling relations. These approximations include: (i) ignoring the asymptotic current sheet width a∞ in Eq. (5); (ii) using an approximate linear growth rate Eq. (15) that ignores the stability threshold; (iii) ignoring the contribution from the lower bound φ0 of the integral in Eq. (22); (iv) assuming the initial or system noise spectrum to be slowly varying compared to the exponential factor in Eq. (34); and (v) replacing the integral in (43) by a Gaussian integral and pushing the bounds to ±∞. Among these approximations, (iv) is an assumption that must be satisfied by the noise spectrum ; the other approximations can be justified a posteriori in the limit of high-S and low-noise.
We now test the analytical scaling relations with the results from numerical solutions of the model equation. For the numerical solutions, we do not employ any approximation except the linear growth rate, where Eq. (14) with ζ = 3/2 is used. Using the approximate linear growth rate is not a significant compromise because the expression is nearly exact; note that the analytical calculation employs a further simplified expression for the linear growth rate, Eq. (15).
To fix ideas, we employ the same setup as in Ref. 9, i.e., with and . We vary the parameters ϵ, χ, and S for cases with either an initial noise or a system noise, with or without the outflow. We integrate the normalized model equation (6) forward in time along characteristics. Because the wavenumber decreases exponentially in time along a characteristic, it is convenient to set the mesh in the wavenumber equispaced with respect to and set the time step . In this way, the wavenumber shifts exactly one grid space in one time step, which significantly simplifies the time integration along characteristics. We perform the time integration with a second-order trapezoidal scheme. For cases with system noise, in each time step the solution is compared with ; we set to for those where . In each time step, we identify the dominant wavenumber and calculate the plasmoid size with Eq. (3), where ξ = 1.5 has been used. The time integration continues until the disruption condition is met. Then we record , and at the disruption time .
The analytical scaling relations involve constants ca, ck, cγ, and cχ. These constants depend on because they depend on φd that maximizes the function ; additionally, ca and cχ also depend on g(φd) and . For , we numerically obtain φd ≈ 0.655, g(φd) ≈ 0.841, and . If the outflow is neglected, i.e., taking the limit , they become φd ≈ 0.784, g(φd) ≈ 0.791, and . These values are used in Eqs. (37), (38), (46), and (48) to obtain the constants ca, ck, cγ, and cχ. The values of these constants for different settings reported in this work are listed in Table I. In the following comparisons, we use the precise analytical scaling relations involving the Lambert function, but the leading order approximate scaling relations are almost as good.
We start with a setup having the initial noise . The scalings for , and with respect to S are shown in Fig. 3 for two cases ϵ = 10−10 and ϵ = 10−20. Here, the symbols denote the scalings obtained from numerical solutions, the dash-dotted lines are the analytical scalings, and the black dashed lines are scalings based on the fastest growing mode of the Sweet-Parker current sheet [i.e., by setting in Eqs. (16) and (17)]. We can see that the analytical scalings accurately agree with the numerically results in the high-S regime. On the other hand, while the Sweet-Parker scaling agrees with in the low to moderately-high S regimes, the Sweet-Parker-based scalings for and considerably depart from the numerical results of and ; the difference is especially substantial when S is low. The discrepancies are due to the stretching effect of outflow that makes the wavelength of the dominant mode significantly longer than that of the fastest growing mode.
Next, we test the analytical scalings for the other two cases: (1) system noise instead of initial noise and (2) ignoring the effects of outflow. Here, we set with ϵ = 10−20. We only present the results for and since they provide more stringent tests than and . The scalings of and with respect to S are shown in Fig. 4; for comparison, we also show results that have been presented in Fig. 3 for cases of initial noise. We again see that the analytical scalings are in excellent agreement with the numerical results in the high-S regime. In that regime, the scalings of are quite similar for all three cases; on the other hand, ignoring the effect of the outflow only slightly affects the scaling of compared to the case with a system noise, whereas is higher for the case with an initial noise. The linear growth rate is higher for the initial noise case because the noise decays during the early period when all the modes are stable; therefore, the disruption occurs at a later time compared to the other two cases. As the current sheet is thinner at a later time, the linear growth rate is higher. Note that the Sweet-Parker-based scalings agree quite well with numerical results at the low-S regime when the outflow is ignored, confirming our earlier statement that the discrepancies are due to the effect of the outflow.
The analytical scaling relations are derived under the assumption that the noise spectrum is slowly varying compared to the exponential factor in Eq. (34). For the cases we have tested so far, this assumption is trivially satisfied with . Now we test the analytical scalings for a power-law system noise , with ϵ = 10−20. Figure 5 shows the scalings of and for the cases χ = 1 and χ = 3. We can see that even for a relatively steep noise spectrum with χ = 3, the analytical scalings remain quite accurate.
In conclusion, we have derived analytical scaling relations for the phenomenological model of plasmoid-mediated disruption of evolving current sheets, including the effects of the reconnection outflow. The analytical scaling relations are derived for two scenarios, one with an initial noise and another with a system noise. The former is useful for comparison with numerical simulations or laboratory experiments where an initial perturbation can be injected; the latter is more suitable for describing the plasmoid instability in natural systems. If the effects of outflow are neglected, and with a proper translation of notations, the scalings obtained from the present model agree with that from previous works based on a principle of least time,27,28 up to the leading order approximation.
The phenomenological model is valid in the linear regime of the plasmoid instability up to the point of current sheet disruption. The time scale td for current sheet disruption is typically on the order of a few Alfvén time τA. Here, we estimate the Alfvén times for various systems of interest. For a large-scale coronal mass ejection (CME), the post-CME current sheet can extend to a length beyond several times that of the solar radius ; the Alfvén speed can be estimated from the speeds of moving blobs in the current sheet, typically in the range of 2 × 105–106m/s.40,41 Assuming L ∼ 109m and VA ∼ 106 m/s, the Alfvén time τA is on the order of 1000 s. For ultraviolet bursts in the solar transition region,42,43 we estimate the current sheet length L ∼ 106 m from forward modeling44 and the Alfvén speed VA ∼ 105 m/s from the Doppler broadening of emission line profiles, yielding τA ∼ 10 s. For laboratory magnetic reconnection experiments,45,46 the Alfvén time τA is typically on the order of microseconds, which is within the temporal resolution of in situ measurements.47 Thus, it may be possible to test some of the predictions from the phenomenological model with future generations of laboratory experiments.48
Although the present study focuses on resistive MHD, the methodology is quite general. The analysis could be generalized to include other effects such as viscosity, the Hall effect, or ambipolar diffusion from partially ionized plasmas. It is also important to further test the predictions of the phenomenological model with full resistive MHD simulations, especially in the high-S regime. While the present analytical calculation is limited to the special class of evolving current sheets where the upstream magnetic field remains constant, the phenomenological model could also describe cases where the upstream magnetic field evolves in time, e.g., as in Refs. 49 and 50. These directions will be pursued in the future.
We thank the anonymous referee for many constructive suggestions. This work was supported by the National Science Foundation, Grant Nos. AGS-1338944 and AGS-1460169; the Department of Energy, Grant No. DE-SC0016470; and the National Aeronautics and Space Administration, Grant No. 80NSSC18K1285. A. Bhattacharjee would like to acknowledge the hospitality of the Flatiron Institute, supported by the Simons Foundation.
It always requires some “seed” noise to trigger the plasmoid instability. In MHD simulations that tend to be clean (i.e., with very low system noise due to discretization and round-off errors), a seed noise is usually provided by adding a random noise in the initial condition; this is an example of initial noise. In contrast, particle-in-cell simulations typically do not require an initial noise to trigger the plasmoid instability because they are inherently noisy, but an initial noise can be seeded as well. The random noise in particle-in-cell simulations is an example of system noise. The same applies to laboratory and natural systems. The plasmoid instability in such systems is usually triggered by the system noise, but an initial noise may also be introduced in laboratory experiments.
This assumption is not valid if is narrowband. In that case, we have to locate the maximum of the full .
This can be seen as follows. Note that the dominant wavenumber scales in the same way as the fastest-growing wavenumber at the disruption time, where the latter is at the transition between the small- and the large- regimes. Because along a characteristic and , the wavenumber is larger than at an earlier time with , which places the mode in the small- regime.