Optimal Energy Growth in Current Sheets

Abstract

In this article, we investigate the possibility of transient growth in the linear perturbation of current sheets. The resistive magnetohydrodynamics operator for a background field consisting of a current sheet is non-normal, meaning that associated eigenvalues and eigenmodes can be very sensitive to perturbation. In a linear stability analysis of a tearing current sheet, we show that modes that are damped as $t\rightarrow \infty$ can produce transient energy growth, contributing faster growth rates and higher energy attainment (within a fixed finite time) than the unstable tearing mode found from normal-mode analysis. We determine the transient growth for tearing-stable and tearing-unstable regimes and discuss the consequences of our results for processes in the solar atmosphere, such as flares and coronal heating. Our results have significant potential impact on how fast current sheets can be disrupted. In particular, transient energy growth due to (asymptotically) damped modes may lead to accelerated current sheet thinning and, hence, a faster onset of the plasmoid instability, compared to the rate determined by the tearing mode alone.

Appendix

Throughout this article, we have performed calculations with boundary conditions $u=b=0$ at $x=\pm d$. This has been done so that our results can be easily compared to other works and to nonlinear simulations, which typically use such boundary conditions. Since the tearing instability develops in a boundary layer near $x=0$, the precise nature of the boundary conditions should not play a strong role in the onset of the instability. To illustrate this, we solve the discrete form of Equation 23, with $S=1000$ and $k=0.5$, in the half-plane and compare the resulting spectrum to that in Figure 1a.

Anticipating a symmetric solution in $\boldsymbol{b}$ and an antisymmetric solution in $\boldsymbol{u}$ about $x=0$, we set the boundary conditions at $x=0$ to be

$$ u=\frac{\mathrm{d}b}{\mathrm{d}x}=0. $$

(61)

As $x\rightarrow \infty$, we set

$$ u=b=0. $$

(62)

In order to represent this boundary numerically, we consider a large domain denoted by $0\le x\le x_{\mathrm{max}}$. In order to expand the variables using Chebyshev polynomials, we need to map our coordinates to the domain $-1\le y \le 1$. This is achieved through

$$ x = a\frac{1+y}{b-y}, $$

(63)

where

$$ a = \frac{x_{\mathrm{max}}x_{i}}{x_{\mathrm{max}}-2x_{i}} \quad \mbox{and} \quad b=1+\frac{2a}{x_{\mathrm{max}}}. $$

(64)

This mapping clusters the grid points near the boundary layer at $x=0$ and places half of the grid points in the region $0\le x\le x_{i}$ (Hanifi, Schmid, and Henningson, 1996). In this example, we take $x_{\mathrm{max}}=100$ and $x_{i}=15$. The resulting spectrum is displayed in Figure 6.

Figure 6

Spectrum of the discrete eigenvalue problem with half-plane boundary conditions for $S=1000$, $k=0.5$.

By inspection, the comparison with Figure 1a yields few differences. The eigenvalue corresponding to the tearing mode now has a value ${\approx}\, 0.0111$, which is still similar to that calculated for the other boundary conditions. Two isolated eigenvalues near $\Im (\omega )=0$ in Figure 1a are now pushed nearer the main branches in Figure 6. Apart from these minor differences, the spectra calculated from different boundary conditions are very similar. This result suggests that the exact form of boundary conditions, assuming they do not interfere dynamically with the boundary layer at $x=0$, will not radically change the behavior of the onset of the tearing instability.