Marginal Stability of Sweet–Parker Type Current Sheets at Low Lundquist Numbers

Magnetic reconnection is a phenomenon where magnetic energy is converted into the thermal and kinetic energy of the plasma via an induction electric field, leading to a reconfigured magnetic field topology. The classic stationary reconnection model was developed by Sweet and Parker (SP; Parker 1957; Sweet 1958), who studied magnetic field annihilation in a two-dimensional current layer with a macroscopic half-length L. It predicts a reconnection rate $R\sim {S}^{-1/2}$ where S is the Lundquist number defined as $S={{LV}}_{\mathrm{Au}}/\eta$ (V_Au and η are the upstream Alfvén speed and the magnetic diffusivity). For current sheets in space plasmas, S is usually very large (e.g., ∼10¹³ in the solar corona), and the typical timescales are much longer than the duration of explosive events such as flares: thus, even though the inverse aspect ratio of the current sheet is small, $a/L\simeq {S}^{-1/2}$ (a is the half thickness of the current sheet), the SP model cannot explain explosive reconnection properly.

However, simulations by Biskamp (1986) showed that the SP stationary reconnecting current sheet becomes unstable to an extremely fast super-tearing, or plasmoid instability, once a critical value for the Lundquist number ${10}^{4}\leqslant {S}_{c}\leqslant {10}^{5}$ is exceeded. Loureiro et al. (2007) pointed out (after Tajima & Shibata 2002) that the maximum growth rate of the instability scales as $\sim {S}^{1/4}$. The positive scaling of the instability growth rate with Lundquist number, and its divergence in the ideal limit, led Pucci & Velli (2014) to point out that the only conclusion of such stability studies should be that large-S SP current sheets should not form in nature in the first place. They identified a critical aspect ratio scaling $L/a\sim {S}^{1/3}$ above which current sheets become unstable even in the limit of ideal MHD. The critical aspect ratio scaling separates current sheets for which the tearing instability diverges, from those in which the growth rate of the tearing mode goes to zero as $S\to \infty$. At the critical scaling, the tearing instability becomes ideal—the growth rate becomes independent of the Lundquist number. Resistive MHD simulations by Tenerani et al. (2015) and Landi et al. (2015) have confirmed the existence and relevance of the critical aspect ratio even for nonlinear simulations.

The works mentioned above all focus on large Lundquist numbers (S ≥ 10⁵), though as mentioned above the early MHD simulations suggested that at S ≤ 10⁴ the current sheet is actually stable and laminar (Biskamp 1986). Bulanov et al. (1978) proposed that, unlike static current sheets, which are always unstable (albeit to the very slow, for macroscopic current layers, tearing instability), the inhomogeneous outflow of the SP configuration can have a stabilizing effect leading to the Lundquist number threshold for the current sheet stability. Biskamp (2005) further analyzed Bulanov's model and estimated roughly the threshold for the aspect ratio L/a to be ∼10², corresponding to a Lundquist number S ∼ 10⁴. However, Loureiro et al. (2007) claimed that the outflow only plays a mildly stabilizing role. Indeed, it was further suggested (Loureiro et al. 2013) that stabilization might be provided by a more mathematical criterion, namely that in order for the tearing instability to develop in an SP current sheet, the inner singular layer that tearing develops should be significantly narrower, i.e., by about a factor of 3, than the overall current sheet thickness. Remark that this criterion might be important in invalidating the classical asymptotic expansions (Furth et al. 1963) with which the analytical approximation of the dispersion relation for resistive modes was originally derived, but arguably has no real physical meaning: indeed, as we will discuss further in the conclusions, it cannot justify the observed SP stabilization.

More recently, the 2D linear MHD simulation by Ni et al. (2010) confirmed that, though the effect of flows is negligible when S is large, its importance increases with decreasing S. In their simulations, initial random perturbations grow linearly at first but then saturate. They determine a critical Lundquist number S ∼ 2000 at which the random perturbation only grows by a factor of 5. The simulation by Ni et al. (2010) is periodic in the outflow direction but employs a "viscous buffer region" at the outer boundaries where viscosity is very large to damp out perturbations: the need to have vanishing boundary conditions and associated damping regions might therefore influence the stability criterion significantly. Indeed, realistic background configurations are periodic neither in the outflow nor inflow directions, rather the flow is accelerated into a jet diverging from the sheet at the background Alfvén speed.

In this study, we carry out linear MHD simulations and linear stability analysis of 2D Sweet–Parker type current sheets at low Lundquist numbers (S ≤ 10⁴). We use open-boundary conditions in both the inflow and outflow directions in the simulations. Our result shows that the inhomogeneous outflow stretches the magnetic islands forming and growing in the current sheet, while at the same time evacuating them from the simulation domain. Both stretching and evacuation play a stabilizing role. In addition, the stretching effect changes the linear instability from the exponential behavior of the static instability, to one in which growth is only exponential over a limited time interval. We confirm that the overall growth of fluctuations is limited, the reason being that the effective wave number of unstable perturbations decreases over time limiting the overall period of instability.

The following section describes our numerical method. Section 3 describes the numerical results. Sections 4 and 5 then describe SP stability, while Section 6 shows numerical results for a different background configuration. Section 7 describes the relevant outlook of our results.

The governing equations for the resistive stability of SP current sheets are given by the linearized resistive MHD equations (and the adiabatic equation for closure—this decouples the dynamic instability from potential thermodynamic effects):

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rho }_{1}}{\partial t}+{{\boldsymbol{u}}}_{0}\cdot {\rm{\nabla }}{\rho }_{1}+{\rho }_{0}{\rm{\nabla }}\cdot {{\boldsymbol{u}}}_{1}+{{\boldsymbol{u}}}_{1}\cdot {\rm{\nabla }}{\rho }_{0}+{\rho }_{1}{\rm{\nabla }}\cdot {{\boldsymbol{u}}}_{0}=0\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {{\boldsymbol{u}}}_{1}}{\partial t}+{{\boldsymbol{u}}}_{0}\cdot {\rm{\nabla }}{{\boldsymbol{u}}}_{1}+{{\boldsymbol{u}}}_{1}\cdot {\rm{\nabla }}{{\boldsymbol{u}}}_{0}+\displaystyle \frac{1}{4\pi {\rho }_{0}}\\ &&\quad \times [{\rm{\nabla }}{{\boldsymbol{b}}}_{1}\cdot {{\boldsymbol{B}}}_{0}-{{\boldsymbol{B}}}_{0}\cdot {\rm{\nabla }}{{\boldsymbol{b}}}_{1}+{\rm{\nabla }}{{\boldsymbol{B}}}_{0}\cdot {{\boldsymbol{b}}}_{1}-{{\boldsymbol{b}}}_{1}\cdot {\rm{\nabla }}{{\boldsymbol{B}}}_{0}]\\ &&\quad +\displaystyle \frac{{\rho }_{1}}{{\rho }_{0}}{{\boldsymbol{u}}}_{0}\cdot {\rm{\nabla }}{{\boldsymbol{u}}}_{0}=-\displaystyle \frac{1}{{\rho }_{0}}{\rm{\nabla }}{p}_{1}\end{eqnarray} \tag{ 1b }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\phi }_{1}}{\partial t}-{{\boldsymbol{u}}}_{0}\times {{\boldsymbol{b}}}_{1}-{{\boldsymbol{u}}}_{1}\times {{\boldsymbol{B}}}_{0}-\eta {{\rm{\nabla }}}^{2}{\phi }_{1}=0\end{eqnarray} \tag{ 1c }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {T}_{1}}{\partial t}+{{\boldsymbol{u}}}_{0}\cdot {\rm{\nabla }}{T}_{1}+(\kappa -1)({\rm{\nabla }}\cdot {{\boldsymbol{u}}}_{1}){T}_{0}\\ &&\quad +\,{{\boldsymbol{u}}}_{1}\cdot {\rm{\nabla }}{T}_{0}+(\kappa -1)({\rm{\nabla }}\cdot {{\boldsymbol{u}}}_{0}){T}_{1}=0.\end{eqnarray} \tag{ 1d }$$

The variables with subscript "0" are the stationary background fields and those with subscript "1" are the perturbed fields that are evolved by the code. ${p}_{1}={\rho }_{0}{T}_{1}+{\rho }_{1}{T}_{0}$ is the perturbed pressure, ρ and T are density and temperature, respectively, and ${\boldsymbol{u}}$ is the velocity and the adiabatic index κ = 5/3. ϕ₁ is the perturbed magnetic flux function from which the perturbed magnetic field ${{\boldsymbol{b}}}_{1}$ is obtained

$$\begin{eqnarray*}&&{{\boldsymbol{b}}}_{1}=\displaystyle \frac{\partial {\phi }_{1}}{\partial y}{\hat{e}}_{x}-\displaystyle \frac{\partial {\phi }_{1}}{\partial x}{\hat{e}}_{y}.\end{eqnarray*}$$

The simulation domain is a 256 × 256 rectangular box for which the x/y axis is along/across the current sheet. As mentioned in the Introduction, the Lundquist number is defined with the half length of the box L:

$$\begin{eqnarray}&&S=\displaystyle \frac{{{LV}}_{\mathrm{Au}}}{\eta }.\end{eqnarray} \tag{ 2 }$$

In our simulations, the length of the box is fixed to be 2 (−1 ≤ x ≤ 1), which means all the lengths are normalized to the half length of the current sheet. As we are studying the Sweet–Parker-type current sheets, the half width of the current sheet scales with S as:

$$\begin{eqnarray}&&\displaystyle \frac{a}{L}={S}^{-\tfrac{1}{2}},\end{eqnarray} \tag{ 3 }$$

so a changes with the Lundquist number in our simulations. As a result, the width of the simulation domain is also changing according to the Lundquist number in order to maintain proper range and resolution in the y direction ($\max (| y| )\sim 10a$). Derivatives are calculated using the sixth order Compact Finite Difference scheme (Lele 1992). At the boundaries, projected characteristics are applied to keep the boundaries open (see Appendix A and Landi et al. 2005). We use a fourth order Runge–Kutta method to advance in time.

The background fields are given by

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{0}={B}_{0}(y){\hat{e}}_{x}\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&{{\boldsymbol{u}}}_{0}={\rm{\Gamma }}x{\hat{e}}_{x}-{\rm{\Gamma }}y{\hat{e}}_{y},\end{eqnarray} \tag{ 4b }$$

where Γ is a constant controlling how fast the outflow is accelerated. For Sweet–Parker-type current sheets, the outflow speed is the upstream Alfvén speed so we have

$$\begin{eqnarray}&&{\rm{\Gamma }}=\displaystyle \frac{{V}_{\mathrm{Au}}}{L}={\tau }_{{\rm{A}}}^{-1},\end{eqnarray} \tag{ 5 }$$

where τ_A is the Alfvén crossing time of the current sheet. The plot of ${{\boldsymbol{u}}}_{0}$ is shown in the left panel of Figure 1. ρ₀ is uniform and set as 1/4π, so the local Alfvén speed is simply

$$\begin{eqnarray}&&{V}_{{\rm{A}}}(y)={B}_{0}(y).\end{eqnarray} \tag{ 6 }$$

${T}_{0}(x,y)$ is calculated by

$$\begin{eqnarray}&&{T}_{0}(x,y)={\bar{T}}_{0}-\displaystyle \frac{1}{2}({u}_{0x}^{2}+{u}_{0y}^{2}+{B}_{0}^{2}),\end{eqnarray} \tag{ 7 }$$

so that the zeroth order momentum equation is satisfied. ${\bar{T}}_{0}$ is a constant and set to be 2 in the simulations. Note that self-consistency requires that ${\rm{\nabla }}\times ({{\boldsymbol{u}}}_{0}\cdot {\rm{\nabla }}{{\boldsymbol{u}}}_{0}-\tfrac{1}{4\pi {\rho }_{0}}{{\boldsymbol{B}}}_{0}\cdot {\rm{\nabla }}{{\boldsymbol{B}}}_{0})=0$, which is satisfied by Equation (4).

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The self-consistent equilibrium magnetic field ${B}_{0}(y)$ is

$$\begin{eqnarray}&&{B}_{0}(y)=\displaystyle \frac{{\bar{B}}_{0}}{0.54}\exp \left[-{\left(\displaystyle \frac{y}{\sqrt{2}a}\right)}^{2}\right]{\int }_{0}^{y/\sqrt{2}a}{e}^{{s}^{2}}{ds},\end{eqnarray} \tag{ 8 }$$

which is shown as the blue curve in the right panel of Figure 1. For this equilibrium, the amplitude of B₀ has a maximum and then decreases toward 0 far from the center of the sheet. In other words, because of the background inflow, the magnetic field is concentrated near the midplane of the current sheet and we use the peak value ${\bar{B}}_{0}$ (set as 1) as the characteristic ("upstream") magnetic field in this case. In Sections 3–5, we adopt this equilibrium field for the simulations and the linear stability analysis. In Section 6, we also show simulation results based on the widely used Harris current sheet field

$$\begin{eqnarray}&&{B}_{0}(y)={\bar{B}}_{0}\tanh \left(\displaystyle \frac{y}{a}\right),\end{eqnarray} \tag{ 9 }$$

which is shown as the orange curve in the right panel of Figure 1, where ${\bar{B}}_{0}$ (set as 1) is the asymptotic value of B₀.

Note that both L and V_Au are fixed to be 1 and thus the Lundquist number is determined solely by the resistivity η in the code. Except for the runs shown in Section 5, we initiate the simulations with white noise, i.e., perturbations of the magnetic flux ϕ₁ uniformly distributed in k space with random phases peaking near the midplane of the current sheet.

In this section, we describe simulation results of the runs with the self-consistent background fields. Figure 2 (left column) shows the evolution of the perturbed magnetic flux function ${\phi }_{1}$ in the run $S={10}^{4}$ (a = 0.01). From the top to bottom panels, we observe that magnetic islands form and grow in amplitude. At the same time, they are stretched along the outflow direction and the islands near the left and right boundaries of the simulation box are ejected out of the domain. The right column of Figure 2 shows the profiles of b_1y and u_1y along the y axis at t = 1.4. As will be shown in Section 4, these are essentially identical to the eigenfunctions, which may be calculated from the linear theory by incorporating the expanding-wavelength assumption

$$\begin{eqnarray}&&k(t)={k}_{0}\exp (-{\rm{\Gamma }}t).\end{eqnarray} \tag{ 10 }$$

This ansatz was first proposed by Bulanov et al. (1978) to transform the problem into a 1D eigenvalue calculation for each value of k for the tearing mode in a current sheet with a linearly accelerating outflow ${u}_{0x}={\rm{\Gamma }}x$. Note, however, that in the 2D current sheet, it is impossible to strictly define a purely monochromatic wave because of the lack of periodicity along x. In any case, the 1D assumption takes the stretching effect of the background flow properly into account: imagine two points separated by one wavelength $\lambda (t)$ along the flow

$$\begin{eqnarray}&&{x}_{2}(t)={x}_{1}(t)+\lambda (t).\end{eqnarray} \tag{ 11 }$$

After time ${dt}$, the distance between the two points becomes

$$\begin{eqnarray}\lambda (t+{dt}) & = & {x}_{2}-{x}_{1}+[{u}_{0x}({x}_{2})-{u}_{0x}({x}_{1})]{dt}\\ & = & \lambda (t)+\displaystyle \frac{{{du}}_{0x}}{{dx}}\lambda (t){dt},\end{eqnarray} \tag{ 12 }$$

which means

$$\begin{eqnarray}&&\displaystyle \frac{1}{\lambda }\displaystyle \frac{d\lambda }{{dt}}=\displaystyle \frac{{{du}}_{0x}}{{dx}}\end{eqnarray} \tag{ 13 }$$

and we easily get Equation (10). The result shown in Figure 2 is qualitatively consistent with the time-dependent wave number assumption: we will discuss this in more detail in Section 4.

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We calculate the perturbed magnetic and kinetic energies inside the simulation domain

$$\begin{eqnarray}&&{E}_{k}=\int \displaystyle \frac{1}{2}({u}_{1x}^{2}+{u}_{1y}^{2}),\quad {E}_{b}=\int \displaystyle \frac{1}{2}({b}_{1x}^{2}+{b}_{1y}^{2})\end{eqnarray} \tag{ 14 }$$

as functions of time. These are shown in Figure 3: from top left to bottom right, the panels display runs for $S={10}^{4},{10}^{3},300$, and 100 respectively. One immediate result from Figure 3 is that the growth rate of the energy in the perturbed fields decreases with decreasing Lundquist number. At S ∼ 100, the perturbed energy decreases with time. Another feature observed in Figure 3 is that the energy in the x-component of the fields dominates over the y-component energy. This is very clear for runs $S={10}^{4},{10}^{3}$, and 300 where the ratios ${E}_{{kx}}/{E}_{{ky}}$ and ${E}_{{bx}}/{E}_{{by}}$ tend to increase with time. This phenomenon can also be explained qualitatively by the stretching effect of the inhomogeneous outflow. Consider the first order magnetic field ${{\boldsymbol{b}}}_{1}$, which must be divergence free:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {b}_{1x}}{\partial x}+\displaystyle \frac{\partial {b}_{1y}}{\partial y}=0.\end{eqnarray} \tag{ 15 }$$

If the perturbations are approximated by a mode with a time-dependent wave number (Equation (10)), we can estimate

$$\begin{eqnarray}&&\displaystyle \frac{\partial {b}_{1x}}{\partial x}\sim {k}_{0}{b}_{1x}\exp (-{\rm{\Gamma }}t)\end{eqnarray} \tag{ 16 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{\partial {b}_{1y}}{\partial y}\sim \displaystyle \frac{{b}_{1y}}{\delta },\end{eqnarray} \tag{ 17 }$$

where δ is the thickness of the inner layer of the tearing mode (Loureiro et al. 2007) and is approximately constant with time. We then get

$$\begin{eqnarray}&&\displaystyle \frac{{b}_{1y}}{{b}_{1x}}\sim ({k}_{0}\delta )\exp (-{\rm{\Gamma }}t),\end{eqnarray} \tag{ 18 }$$

which means that the growth of b_1y is slower than that of b_1x. We can apply the same analysis to ${{\boldsymbol{u}}}_{1}$, assuming that the incompressible modes dominate.

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In practice, b_1y, as the reconnecting component of the field, is the more interesting quantity. Figure 4 shows the growth of b_1y in several runs. The upper panel shows the amplitude of b_1y as a function of time for runs with different S. Here we use the maximum of $| {b}_{1y}|$ inside the whole simulation domain to represent the amplitude of b_1y. For S < 300, $| {b}_{1y}|$ does not grow with time. For the runs with S > 300, $| {b}_{1y}|$ grows after an initial transient phase and then the growth gradually slows down until reaching a maximum. The two dots on each curve mark the start and the end of the growth phase. The saturation time corresponds roughly to the time when most of the magnetic islands are ejected out. We use the ratio between the amplitudes of b_1y at the two points to measure the total growth of $| {b}_{1y}|$ during a simulation, which is shown as the blue dots in the lower panel of Figure 4 as a function of Lundquist number. The orange dots in the same panel are the total growth of b_1y estimated through the linear theory, which will be discussed in the next section. At S = 1000, $| {b}_{1y}|$ grows by a factor of 17, while at S = 10⁴ it grows by a factor of ∼3.5 × 10⁵. If we arbitrarily choose the criterion that $| {b}_{1y}|$ grows by a factor of 10² in order to perturb the current sheet strongly, the critical Lundquist number needs to be ∼2000. Here "strongly perturb the current sheet" means that the amplitude of the perturbation becomes large enough to trigger a fast nonlinear evolution (Tenerani et al. 2015, 2016).

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If we use the same background flow as that used in the simulations (Equation 4(b)) and include the expanding effect of the outflow on wavelengths by using a time-dependent wave vector such as that given by Equation (10), assuming that v and b are functions of y only (rigorously they should be functions of both y and t), of the form

$$\begin{eqnarray}&&\left(\begin{array}{c}{u}_{1y}\\ {b}_{1y}\end{array}\right)=\left(\begin{array}{c}-{iv}(y)\\ b(y)\end{array}\right)\exp \left[{ik}(t)x+{\int }_{0}^{t}\gamma (k({t}^{{\prime} })){{dt}}^{{\prime} }\right],\end{eqnarray} \tag{ 19 }$$

we can reduce Equation (1) to a set of ordinary differential equations

$$\begin{eqnarray}&&y({v}^{\prime\prime\prime }-{k}^{2}{v}^{{\prime} })-(\gamma +1){v}^{{\prime\prime} }+(\gamma -1){k}^{2}v\\ &&\quad =\,k\displaystyle \frac{L}{a}[{B}_{0}(y)({b}^{{\prime\prime} }-{k}^{2}b)-{B}_{0}^{\prime\prime} (y)b]\end{eqnarray} \tag{ 20a }$$

$$\begin{eqnarray}&&(\gamma +1)b-{{yb}}^{{\prime} }=k\displaystyle \frac{L}{a}{B}_{0}(y)v+\displaystyle \frac{1}{S}\,\displaystyle \frac{{L}^{2}}{{a}^{2}}({b}^{{\prime\prime} }-{k}^{2}b).\end{eqnarray} \tag{ 20b }$$

In deriving Equation (20), we have also assumed incompressibility and adopted the following normalizations: length to the half thickness of the current sheet a, magnetic field to the upstream magnetic field ${\bar{B}}_{0}$, speed to the upstream Alfvén speed V_Au, and time to Alfvén time ${\tau }_{{\rm{A}}}=L/{V}_{\mathrm{Au}}$, where L is the half length of the current sheet. S is defined as ${{LV}}_{\mathrm{Au}}/\eta$ and we study Sweet–Parker-type current sheets so that $a/L={S}^{-1/2}$. Equation (20) is a 1D eigenvalue problem with the boundary conditions that v and b vanish at infinity. We numerically solve this eigenvalue problem (see Appendix B) and present the result in Figure 5. The upper panel of Figure 5 is the dispersion relation γ (k). The shape of the γ (k) curves is similar to that of the dispersion relation of the classic tearing mode: there is a fastest growing mode k_f for each Lundquist number and the growth rate decreases for larger or smaller k. Unlike the tearing mode in static current sheets, whose growth rate is never negative, now part of the γ(k) curve goes below the γ = 0 line. For S ≲ 70 the whole curve is below γ = 0, which means that in this configuration of the background fields, the tearing mode does not grow at all for S ≲ 70.

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The stabilizing effect of the background flow can be understood by looking at Equation (20). In Equation (20), the most important term that determines γ is $(\gamma +1){v}^{{\prime\prime} }$. In the same equation for the tearing mode inside a static current sheet, the term becomes $\gamma {v}^{{\prime\prime} }$. That is to say

$$\begin{eqnarray}&&\gamma \sim {\gamma }_{s}-1,\end{eqnarray} \tag{ 21 }$$

where γ is the growth rate with the presence of flow and γ_s is the growth rate without the flow. The difference 1(Γ) comes from the time-derivative of k in deriving Equation (20). This means that in the current sheet with the background flow, stretching of the magnetic islands due to the inhomogeneous outflow slows down their growth. Compared with the equations derived by Bulanov et al. (1978; who take into account only the outflow), Equation (20) displays two major differences: (1) the coefficient in front of v'' changes from γ + 2 to γ + 1. (2) y-convection terms appear ($y({v}^{\prime\prime\prime }-{k}^{2}{v}^{{\prime} })$ and ${{yb}}^{{\prime} }$). In Bulanov's equation, the constant 2(Γ) consists of one Γ, which comes from dk/dt (expansion of the wavelength) and another one Γ coming from ${{du}}_{0x}/{dx}$ (expansion of the fluid volume) so their model is even more stable than Equation (20). However, when the inflow is included, the ${{du}}_{0x}/{dx}$ is canceled out by ${{du}}_{0y}/{dy}$ because of incompressibility. In fact, the background configuration considered by Bulanov et al. (1978) is not self-consistent because their ${{\boldsymbol{u}}}_{0}={\rm{\Gamma }}x{\hat{e}}_{x}$ is compressible, while they still assumed a uniform density. The newly included y-convection terms, especially the term $y({v}^{\prime\prime\prime }-{k}^{2}{v}^{{\prime} })$, though not important at large S, affects the calculated γ at low S (≲10³) when $y({v}^{\prime\prime\prime }-{k}^{2}{v}^{{\prime} })$ becomes comparable to $(\gamma +1){v}^{{\prime\prime} }$. We also solved Equation (20) without the two y-convection terms (not shown here) and it turns out that the growth rate becomes larger, i.e., the y-convection terms somewhat contribute to the stabilization of the current sheet. But, as will be discussed further below, at very low S the linear analysis method becomes invalid so we are unable to evaluate exactly how much stabilization the y-convection brings.

In the middle panel of Figure 5, we compare the eigenfunctions calculated through Equation (20) with the profiles taken from the simulation for $S={10}^{4}$. The blue lines are the calculated eigenfunctions at ka = 0.15, i.e., the wavelength λ ≈ 0.4 and the orange lines are u_1y and b_1y along the y axis taken from the S = 10⁴ run at t = 1.4 (right column of Figure 2) when the wavelength of the dominant mode is ≈0.4. The amplitudes of the eigenfunctions and the cuts from the simulation are adjusted to be the same. We can see that the shapes of the eigenfunctions by the linear theory are very close to the profiles taken from the linear simulation, confirming the validity of the linear theory in this case.

Based on the linear theory, we are able to explain the saturation phenomenon observed in Figure 4. For a mode with wave number k(t) at time t, the instantaneous growth rate is $\gamma (k(t))$. As time increases, k(t) decreases and thus the instantaneous growth rate moves along the γ–k curve toward the left. After k(t) passes the fastest growing wave number k_f, the growth rate starts to decrease and eventually goes to 0 (even below 0). To verify this theory, we first estimate the dominant mode, i.e., the mode whose amplitude is the largest, at a certain time and compare it with the simulation result. We assume that the initial perturbations are uniformly distributed in k-space. At time t, the wave number $k=k(t)$ corresponds to the initial wave number ${k}_{0}=k(t){e}^{t}$ (in our normalization Γ = 1). At any time $0\leqslant {t}^{{\prime} }\leqslant t$ this mode corresponds to

$$\begin{eqnarray}&&{k}^{{\prime} }={k}_{0}{e}^{-{t}^{{\prime} }}=k(t){e}^{t-{t}^{{\prime} }},\end{eqnarray} \tag{ 22 }$$

and the instantaneous growth rate at t' is

$$\begin{eqnarray}&&{\gamma }^{{\prime} }=\gamma ({k}^{{\prime} })=\gamma [k(t){e}^{t-{t}^{{\prime} }}].\end{eqnarray} \tag{ 23 }$$

In order to get the amplitude of the mode k(t), we integrate along the $\gamma -k$ curve from k₀ to k(t):

$$\begin{eqnarray}&&| {b}_{1y}| (t,k(t))=\exp \left[{\int }_{0}^{t}\gamma (k\,{e}^{t-{t}^{{\prime} }}){{dt}}^{{\prime} }\right]\end{eqnarray} \tag{ 24 }$$

and then we are able to find the wave number k_d(t) whose amplitude is the largest. In the left panel of Figure 6, the orange curve shows the estimated dominant wave number through Equation (24) for $S={10}^{4}$ and the blue dots show the dominant wave number calculated from the simulation S = 10⁴. In analyzing the simulation result, we first cut ${b}_{1y}(x,y,t)$ along the x axis, i.e., the midplane of the current sheet, and then multiply it by a Tukey window to remove the effect of nonperiodicity along x before we apply Fourier transform to it. The steps in the blue dots are due to the resolution in k space when we do Fourier transform. We shift the orange curve by Δt = 0.1 because of the initial transient phase in the simulation. We see that the curve and the dots almost overlap for t ≳ 1.0. The large difference between them for t ≲ 0.5 may be a result of the initial transient phase in the simulation. As we have the amplitude of b_1y as a function of time from the simulation, we are able to calculate the instantaneous growth rate at a given time by

$$\begin{eqnarray}&&\gamma (t)=\displaystyle \frac{1}{| {b}_{1y}| }\displaystyle \frac{d| {b}_{1y}| }{{dt}}\end{eqnarray} \tag{ 25 }$$

and then relate it with the calculated dominant wave number at the same time to get a γ–k curve, which is shown in the right panel of Figure 6. Again the blue dots are the simulation result and the orange curve is the dispersion relation given by the linear theory (the curve taken from Figure 5). The theory and the simulations agree with each other very well, verifying the physical picture that γ moves along the γ–k curve.

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We estimate the total growth of $| {b}_{1y}|$ using the dispersion relation shown in Figure 5 by doing the integral along the γ–k curves:

$$\begin{eqnarray}&&G(S)=\exp \left[{\int }_{0}^{{t}_{c}}\gamma ({k}_{0}{e}^{-{t}^{{\prime} }}| S){{dt}}^{{\prime} }\right],\end{eqnarray} \tag{ 26 }$$

where t = 0 is at the right zero point of the γ–k curve and t_c is when ${k}_{0}\exp (-{t}_{c})=\pi /L$, i.e., when the wavelength reaches the length of the current sheet (generally very close to the left zero point of the curve). Note that Equation (26) is actually an estimate of the upper limit of the total growth of $| {b}_{1y}|$ because we integrate through the whole positive part of each γ–k curve, thus the estimate is larger than the real growth. The calculated G(S) is shown as the orange dots in the lower panel of Figure 4. We can see that it is in accordance with the simulation, though, a little bit larger and in general the theory result and the simulation result differ by a factor of smaller than 2.

Although the linear stability analysis based on the $k\,={k}_{0}\exp (-{\rm{\Gamma }}t)$ assumption works quite well for S ≳ 10³, we must point out that, for small Lundquist numbers, this assumption becomes invalid because of the small growth rate ($\gamma \lesssim {\tau }_{A}^{-1}$), i.e., when the growth timescale for a magnetic island becomes larger than the timescale for it to be ejected out of the current sheet. Actually, when $\gamma \lt {\rm{\Gamma }}(=\,{\tau }_{{\rm{A}}}^{-1})$, we cannot assume that the functions v and b in Equation (20) are independent of time anymore. Note that in Equation (20) k is a function of time, so mathematically the assumption Equation (19) is not rigorous. Stated otherwise, we are unable to pose a 1D eigenvalue problem for a 2D Sweet–Parker-type current sheet at very small Lundquist numbers. In the lower panel of Figure 5, we compare the simulation result with the eigenfunctions for S = 500 and ka = 0.20. Clearly, the simulation result deviates from the eigenfunctions. There is no doubt that as S becomes smaller the difference between the simulation and the approximate 1D linear theory will become larger. The linear theory predicts a threshold S_c ≈ 70 below which b_1y does not grow while from the upper panel of Figure 4 we see that the threshold is at least ≈300.

In this section, we discuss the stability of the current sheet in terms of initial conditions, i.e., in terms of the initial perturbation. Our simulation is linear so that the amplitude of the initial noise is arbitrary. In Section 3, we chose the criterion for an unstable current sheet to be that $| {b}_{1y}|$ grows by a factor of 10², which gives a critical Lundquist number of ∼2000. However, that is not a universal criterion, because depending on the initial state, we must allow $| {b}_{1y}|$ to grow sufficiently that the sheet is strongly affected, i.e., nonlinear evolution is triggered, at times comparable to their evacuation from the domain.

Because the configuration of the current sheet in this study is fully 2D and the simulation is open-boundary, it is natural to propose that the shape of the initial perturbation also affects the stability. To be more specific, where the initial perturbation is excited on the current sheet is important because the generation location of a magnetic island determines how long the island grows before it is ejected out of the current sheet. Here the location refers to the distance to x = 0, i.e., the location along the outflow. Based on the S = 10⁴ run, we carried out three more runs with identical background configuration but for each run we modulate the initial perturbation by multiplying it with a Gaussian function:

$$\begin{eqnarray}&&f(x| {x}_{c})=\displaystyle \frac{1}{2}\left[\exp \left(-\displaystyle \frac{{(x-{x}_{c})}^{2}}{2{\sigma }^{2}}\right)+\exp \left(-\displaystyle \frac{{(x+{x}_{c})}^{2}}{2{\sigma }^{2}}\right)\right],\end{eqnarray} \tag{ 27 }$$

where x_c = 0.0, 0.4, or 0.8 and σ = 0.1. The three modulation functions are plotted in the left panel of Figure 7. Through the modulation functions, we are able to localize the initial perturbation around $\pm {x}_{c}$. We calculate the amplitudes b_1y as functions of time for each of the three runs and present the results in the right panel of Figure 7. The red dashed line shows the result of the original run without modulation, i.e., the S =10⁴ curve in the upper panel of Figure 4. We see that the curve of the x_c = 0.0 run almost overlaps with the original run, indicating that in the simulation where the initial perturbation is uniformly distributed along the current sheet, the dominant magnetic islands are those generated near the center of the current sheet, as is clearly seen in the left column of Figure 2. In the x_c = 0.4 run, the growth rate of $| {b}_{1y}|$ is almost the same as the x_c = 0.0 run at short times (t ≲ 1.0) but later the growth rapidly stops. As a result, the total growth of $| {b}_{1y}|$ in this run is a factor of ∼1.5 × 10³ rather than the factor 3.5 × 10⁵ of the x_c = 0.0 run. For x_c = 0.8, there is nearly no growth phase for $| {b}_{1y}|$. These results agree with the idea that the evacuation time of the magnetic island is a key factor in determining how much the island can grow. Assuming that a magnetic island is generated at x = x_c, we can estimate the evacuation time of the magnetic island

$$\begin{eqnarray}{\tau }_{e}({x}_{c}) & = & {\displaystyle \int }_{{x}_{c}}^{1}\displaystyle \frac{{dx}}{{u}_{0x}(x)}={\displaystyle \int }_{{x}_{c}}^{1}\displaystyle \frac{{dx}}{x}\\ & = & \mathrm{ln}\left(\displaystyle \frac{1}{{x}_{c}}\right).\end{eqnarray} \tag{ 28 }$$

We then get ${\tau }_{e}(0.4)=0.92$ and ${\tau }_{e}(0.8)=0.22$, which agree with the simulation results.

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We carry out the same simulations as those in Section 3 but change B₀ to the Harris current sheet field (Equation (9)) and present the results in Figures 8 and 9. We also carry out the simplified 1D linear stability analysis with the Harris current sheet field, shown in Figure 10, where the dashed line is the result for a static Harris current sheet and is plotted to show the stabilizing effect of the background flow (see Equation (21) and the discussion in Section 4). Note that we are still assuming the background fields are stationary; though, technically speaking, the velocity field Equation 4(b) and the Harris magnetic field Equation (9) do not satisfy the induction equation. Indeed, the zeroth order fields should vary with time and their evolution will hence affect the evolution of the first order quantities. In this section, however, we ignore the non-self-consistency of the background fields to illustrate the dependence of the current sheet stability on different background magnetic fields. In reality, this implies that our results are physically significant only for growth rates large enough ($\gamma {\tau }_{{\rm{A}}}\gt 1$) where the evolution of the background fields is negeligible.

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Compared with the self-consistent B₀ case, the growth rate with ${B}_{0}(y)={\bar{B}}_{0}\tanh (y/a)$ is smaller. By comparing Figure 8 with Figure 2, we see that the magnetic islands are much thinner with the self-consistent field because of the more concentrated B₀ and that is why the growth rate with the self-consistent B₀ is larger. The growth of $| {b}_{1y}|$ is also much smaller than before: at S = 1000, $| {b}_{1y}|$ grows by a factor of 1.9 (17 for the self-consistent field), while at S = 10⁴ it grows by a factor of 400 (3.5 × 10⁵ for the self-consistent field). If we adopt the same criterion for an unstable current sheet as in Section 3 that $| {b}_{1y}|$ grows by a factor of 10², the critical Lundquist number is at least ∼6000 (2000 for the self-consistent field). Despite of the quantitative differences, the qualitative properties we discuss in Sections 3–5, i.e., the expanding-wavelength and the finite growth time effect, are also applicable here.

In this study, we have explored the tearing instability inside 2D Sweet–Parker type current sheets at low Lundquist numbers using both 2D linear open-boundary MHD simulations and a simplified 1D linear stability analysis. We find that in the linear stability analysis, the assumption of a time-dependent wave number $k={k}_{0}{e}^{-{\rm{\Gamma }}t}$ proposed by Bulanov et al. (1978) works quite well for S ≳ 10³. As predicted by the linear theory, the simulations show that the inhomogeneous background outflow stretches the growing magnetic islands and the stretching slows down the island growth. On the other hand, because of the finite size of the current sheet, the magnetic islands are evacuated out of the current sheet within a finite time (several Alfvén times) and thus they cannot grow infinitely (as would happen in a periodic current sheet). In the simulations initialized with white noise, we clearly observe a saturation phase after the growing phase, corresponding to the time when most of the magnetic islands are ejected out. As a consequence, the excitation location of the initial perturbation is an important factor in determining how much perturbations can grow.

We estimate through the simulations the critical Lundquist number above which the current sheet can be viewed as "strongly perturbed." The result is dependent on the shape of the background magnetic field ${B}_{0}(y)$ we use. Generally, the Harris current sheet field is more stable than the self-consistent background field (Equation (8)). Based on the criterion that $| {b}_{1y}|$ grows by a factor of 10² during the growth phase, the Lundquist number threshold can be ∼2000 or ∼6000 depending on which ${B}_{0}(y)$ is used. We may say that under this specific criterion the threshold S_c is several thousands. However, no universal criterion that is independent of context can be defined, since amplitude and localization of fluctuations in the initial current are fundamental to the subsequent evolution.

We find that the flow structure of the SP sheet adequately accounts for the low S stabilization without the need of other factors, such as those proposed by Loureiro et al. (2013). Their suggested instability threshold, given by

$$\begin{eqnarray}&&\displaystyle \frac{\delta }{a}=f\end{eqnarray} \tag{ 29 }$$

where δ is the thickness of the inner layer, a is the thickness of the current sheet, and f ≃ 1/3 is a constant, yields a critical S compatible with findings of some simulations. On the other hand, the scaling of the stabilization criterion with the current sheet aspect ratio appears to go in the wrong direction. Indeed, consider a current sheet whose aspect ratio is

$$\begin{eqnarray}&&\displaystyle \frac{a}{L}\sim {S}^{-\alpha }.\end{eqnarray} \tag{ 30 }$$

The asymptotic scaling of δ then becomes (see, e.g., Tenerani et al. 2016)

$$\begin{eqnarray}&&\displaystyle \frac{\delta }{a}\sim {S}^{-\tfrac{1}{4}(1-\alpha )}\end{eqnarray} \tag{ 31 }$$

and the threshold Lundquist number S_c from the criterion above would become

$$\begin{eqnarray}&&{S}_{c}={\left(\displaystyle \frac{1}{f}\right)}^{\tfrac{4}{1-\alpha }}.\end{eqnarray} \tag{ 32 }$$

For a Sweet–Parker-type current sheet (α = 1/2) and f = 1/3, Equation (32) gives S_c ≈ 6.5 × 10³, close to 10⁴. However, S_c given by Equation (32) is an increasing function of α, which means that the thinner a current sheet is the more stable it will be, contrary to all numerical and physical intuition. The point is that at a low Lundquist number (<10⁴) asymptotic scalings really do not work in the first place. On the other hand, we have demonstrated that velocity fields do indeed provide the right criteria for stability.

In terms of astrophysical structures, such as, for example, coronal magnetic fields or prominence eruptions, fluctuations are always present, propagating from other coronal regions or from the photosphere. As shown by Tenerani et al. (2016), at very high Lundquist numbers, 2D current sheets tend to disrupt in a quasi-self-similar way, in which case flurrys of smaller scale current sheets are generated iteratively. At each step, the Lundquist number decreases, and finally they will cross the critical S at which further disruption is quenched. However, at those scales where S becomes small enough, kinetic effects, explored in Pucci et al. (2017), become fundamental. We plan to address this question in subsequent works.

We would like to thank Fulvia Pucci for many useful discussions. This research was supported by the NSF-DOE Partnership in Basic Plasma Science and Engineering award n. 1619611 and the NASA Parker Solar Probe Observatory Scientist grant NNX15AF34G. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) (Towns et al. 2014) Comet at the San Diego Supercomputer Center through allocation TG-AST160007. XSEDE is supported by National Science Foundation grant number ACI-1548562.

The linear ideal MHD equation set is a partial differential equation system and can be written in the form:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {{\boldsymbol{U}}}_{1}}{\partial t}+{{\boldsymbol{A}}}_{0}\displaystyle \frac{\partial {{\boldsymbol{U}}}_{1}}{\partial x}+{{\boldsymbol{C}}}_{0}\displaystyle \frac{\partial {{\boldsymbol{U}}}_{1}}{\partial y}+{\boldsymbol{D}}=0,\end{eqnarray} \tag{ 33 }$$

where ${{\boldsymbol{U}}}_{1}$ is the vector of the first order variables:

$$\begin{eqnarray}&&{{\boldsymbol{U}}}_{1}=({\rho }_{1},{u}_{1x},{u}_{1y},{u}_{1z},{b}_{1x},{b}_{1y},{b}_{1z},{T}_{1}).\end{eqnarray} \tag{ 34 }$$

The coefficient matrices ${{\boldsymbol{A}}}_{0}$ and ${{\boldsymbol{C}}}_{0}$ include only the zeroth order quantities and the matrix ${\boldsymbol{D}}={{\boldsymbol{FU}}}_{1}$ contains other terms that do not include any derivatives of the first order variables. Compared with the case of the nonlinear MHD equations, the first three terms on the l.h.s. of Equation (33) are unchanged except that the derivatives are for first order variables only and the coefficients are zeroth order. The matrix ${\boldsymbol{D}}$ now contains more terms than the nonlinear case because all the derivatives of the zeroth order quantities are included in it, e.g., $[{{\boldsymbol{u}}}_{1}\cdot {\rm{\nabla }}{\rho }_{0}+({\rm{\nabla }}\cdot {{\boldsymbol{u}}}_{0}){\rho }_{1}]$ in Equation 1(a). Note that ${\boldsymbol{D}}$ does not affect the projection of characteristics because it does not have any "waves" of ${{\boldsymbol{U}}}_{1}$.

The calculation of the characteristics is almost the same as that shown in Landi et al. (2005). For example, along x, we first diagonalize the matrix ${{\boldsymbol{A}}}_{0}$

$$\begin{eqnarray}&&{{\boldsymbol{A}}}_{0}={\boldsymbol{S}}{\rm{\Lambda }}{S}^{-1},\end{eqnarray} \tag{ 35 }$$

where the diagonal matrix

$$\begin{eqnarray}&&{\rm{\Lambda }}=\mathrm{diag}\{{u}_{0x}+{f}_{x},{u}_{0x}+{a}_{x},{u}_{0x}+{s}_{x},\\ &&\qquad {u}_{0x},{u}_{0x}-{s}_{x},{u}_{0x}-{a}_{x},{u}_{0x}-{f}_{x}\}\end{eqnarray} \tag{ 36 }$$

consists of the speeds of the seven MHD modes (projected along the x direction). u_0x, s_x, a_x, and f_x are the local entropy mode speed, slow magnetosonic wave speed, Alfvén speed, and fast magnetosonic wave speed projected along x. Note that when we calculate the characteristics along x, there is no need to consider b_1x because of the divergence-free condition. Then we are able to decompose ${{\boldsymbol{A}}}_{0}\partial {{\boldsymbol{U}}}_{1}/\partial x$ into the linear combinations of the seven characteristics. The result is quite lengthy and is not shown here, but note that one can write down the result easily by adding subscript "0" to all the variables in the coefficients and subscript "1" to all the variables in the derivatives for the equations shown in the Appendix of Landi et al. (2005).

In order to impose an open-boundary condition on a certain side of the simulation box, we decompose the corresponding derivatives into the linear combination of the characteristics, e.g., decompose the x-derivative terms at the right/left boundaries. For each of the seven characteristics, we decide whether it is propagating inward or outward of the domain by looking at its mode speed. If it is outgoing, we keep this characteristic; otherwise, we set it as 0. By doing this, we ensure that no information is going into the simulation domain and the boundary is "open."

When the resistivity is non-zero, the second order derivative term will cause some reflections at the boundaries. But practically, as the resistivity is quite small, the resistive term does not influence the simulation significantly at the boundaries.

Equation (20) is a fifth order homogeneous boundary value problem (BVP) with a regular singular point y = 0. The widely used methods for BVPs, e.g., the shooting method, usually involve the integrals of the ODEs over the domain. However, for ODEs with singular points such as Equation (20) it is hard to evaluate the integrals. Thus we adopt the method of series-expansion to solve Equation (20). For simplicity, we write Equation (20) in the form

$$\begin{eqnarray}&&{ \mathcal L }(v,b| \gamma ,k,S)=0,\end{eqnarray} \tag{ 37 }$$

where ${ \mathcal L }$ is the linear operator corresponding to the ODE set with $\gamma ,k,S$ (and ${B}_{0}(y)$) as parameters. Our goal is, by imposing proper boundary conditions, to determine the dispersion relation $\gamma (k)$ and the solution v(y) and b(y) simultaneously.

Before discussing the series-expansion method, we would like to first clarify the boundary conditions. The equation is fifth order, so five boundary conditions are needed. We set the domain to be $y\in [0,{L}_{y}]$, where L_y = 17 so that the right boundary is far enough to impose the asymptotic boundary conditions. We expect v to be odd and b to be even in y so that

$$\begin{eqnarray}&&v(0)=0,\quad {b}^{{\prime} }(0)=0.\end{eqnarray} \tag{ 38 }$$

For large enough y, ${B}_{0}(y)$ (Equation (8)) can be approximated by

$$\begin{eqnarray}&&{B}_{0}(y)\approx 1.31\times \displaystyle \frac{1}{y}+O(1{/y}^{3}).\end{eqnarray} \tag{ 39 }$$

Far from the center of the current sheet, the perturbations asymptote to 0. Plugging Equation (39) in Equation (20) and eliminating v, we get a fifth order ODE for b, which by keeping only the fifth and fourth order derivative terms a priori, becomes

$$\begin{eqnarray}&&{b}^{(5)}+{({yb})}^{(4)}=0,\end{eqnarray} \tag{ 40 }$$

which gives the solution

$$\begin{eqnarray}&&b(y)\propto \exp \left(-\displaystyle \frac{1}{2}{y}^{2}\right)\end{eqnarray} \tag{ 41 }$$

and we further get

$$\begin{eqnarray}&&v(y)=\displaystyle \frac{1}{1.31}\times {S}^{-\tfrac{1}{2}}\displaystyle \frac{{k}^{2}+\gamma +2}{k}{yb}(y).\end{eqnarray} \tag{ 42 }$$

The three boundary conditions at the right boundary are then given by

$$\begin{eqnarray}&&{b}^{{\prime} }+{L}_{y}b=0\end{eqnarray} \tag{ 43a }$$

$$\begin{eqnarray}&&{v}^{{\prime} }+\left({L}_{y}-\displaystyle \frac{1}{{L}_{y}}\right)v=0\end{eqnarray} \tag{ 43b }$$

$$\begin{eqnarray}&&v-\displaystyle \frac{1}{1.31}{S}^{-\tfrac{1}{2}}\displaystyle \frac{{k}^{2}+\gamma +2}{k}{L}_{y}b=0.\end{eqnarray} \tag{ 43c }$$

Equations (38) and (43) are the five boundary conditions for the BVP. Note that Equations (43) are not equivalent to open-boundary conditions adopted in the simulations unless ${L}_{y}\to \infty$ where all first order quantities vanish. But, as the simulation box is large enough $\max (| y| )\sim 10a$ and the initial perturbations are localized around y = 0, the difference in the boundary conditions is negligible and we are still allowed to make comparisons between the linear theory and the simulations.

We expand v(y), b(y), and ${B}_{0}(y)$ in power series of y around a point y_c

$$\begin{eqnarray}v(y) & = & \displaystyle \sum _{n=0}^{N}{v}_{n}{(y-{y}_{c})}^{n},\quad b(y)=\displaystyle \sum _{n=0}^{N}{b}_{n}{(y-{y}_{c})}^{n},\\ {B}_{0}(y) & = & \displaystyle \sum _{n=0}^{N}{B}_{n}{(y-{y}_{c})}^{n},\end{eqnarray} \tag{ 44 }$$

where v_n and b_n are quantities to be determined and N = 100 is the cut-off order of the series. B_n are given by the Taylor expansion of ${B}_{0}(y)$ around y_c. Once we get the values of v_n and b_n, the solution v(y) and b(y) can be recovered. We plug Equation (44) into Equation (20) and get the recurrence relations by matching the coefficients in front of ${(y\mbox{--}{y}_{c})}^{n}$:

$$\begin{eqnarray}&&{y}_{c}(n+3)(n+2)(n+1){v}_{n+3}=(n+2)(n+1)\\ &&\quad \times (\gamma +1-n){v}_{n+2}+{k}^{2}{y}_{c}(n+1){v}_{n+1}\\ &&\quad -(\gamma -1-n){k}^{2}{v}_{n}+{{kS}}^{\tfrac{1}{2}}\displaystyle \sum _{l=0}^{n}[(l+2)(l+1){B}_{n-l}{b}_{l+2}\\ &&\quad -{k}^{2}{B}_{n-l}{b}_{l}-(n-l+2)(n-l+1){B}_{n-l+2}{b}_{l}]\end{eqnarray} \tag{ 45a }$$

$$\begin{eqnarray}(n+2)(n+1){b}_{n+2} & = & (\gamma +1-n+{k}^{2}){b}_{n}-{y}_{c}(n+1){b}_{n+1}\\ & & -{{kS}}^{\tfrac{1}{2}}\displaystyle \sum _{l=0}^{n}{B}_{n-l}{v}_{l}.\end{eqnarray} \tag{ 45b }$$

From Equation (45), it is obvious that there are only five free parameters (${v}_{0},{v}_{1},{v}_{2},{b}_{0},{b}_{1}$): all the v_n for $n\geqslant 3$ and b_n for n ≥ 2 can be expressed in the five free parameters and so for v(y) and b(y):

$$\begin{eqnarray}v(y) & = & v(y| {v}_{0},{v}_{1},{v}_{2},{b}_{0},{b}_{1}),\\ b(y) & = & b(y| {v}_{0},{v}_{1},{v}_{2},{b}_{0},{b}_{1}).\end{eqnarray} \tag{ 46 }$$

Then the five boundary conditions can be written as a linear algebraic equation set

$$\begin{eqnarray}&&{\boldsymbol{M}}(\gamma ,k,S){\boldsymbol{V}}=0,\end{eqnarray} \tag{ 47 }$$

where ${\boldsymbol{V}}={({v}_{0},{v}_{1},{v}_{2},{b}_{0},{b}_{1})}^{T}$ and ${\boldsymbol{M}}$ is a matrix, which is a function of γ, k, S. In order to get a nontrivial solution, we need

$$\begin{eqnarray}&&\det ({\boldsymbol{M}})=0,\end{eqnarray} \tag{ 48 }$$

which gives the dispersion relation $\gamma (k,S)$ and then we get v(y) and b(y).

In practice, we must do the expansion at multiple points (y_c) to ensure convergence of the series. If we use N_c points to expand the functions, we need $5{N}_{c}$ equations to determine the solution. Apart from the five boundary conditions Equations (38) and (43), the other $5({N}_{c}-1)$ equations come from the continuities of v(y), ${v}^{{\prime} }(y)$, ${v}^{{\prime\prime} }(y)$, b(y), ${b}^{{\prime} }(y)$ at ${N}_{c}-1$ junction points (y_m), each of which lies between two neighboring expansion points. In solving Equation (20), we use 11 expansion points.