3.1. Tearing versus non-tearing modes

The growth of the tearing mode leads to the development of magnetic islands. For this to happen, the normal magnetic field $B_{x}$ must not be zero at the mode resonance surface at $x=x_{0}$ . At the same time, inflow must change the sign at $x=x_{0}$ and thus $v_{x}(x_{0})=0$ . Otherwise a magnetic island will not develop.

We define an unstable mode to be a tearing mode if (1)  $v_{x}$ is an odd function of $x$ about the mode resonance surface $x=x_{0}\leqslant L$ (inside the ECS) so that $v_{x}(x_{0})=0$ and (2)  $b_{x}(x_{0})\neq 0$ . All other modes are considered to be of the non-tearing type. Note that the free energy source for all the unstable modes (both tearing and non-tearing) is the electron shear flow, which is equivalent to electric current for stationary ions.

Applying the tearing mode conditions, viz., $v_{x}(x_{0})=0$ and $b_{x}(x_{0})\neq 0$ , in (2.2) and (2.3), we obtain the necessary conditions for the existence of tearing eigenfunctions.

where we have used $[\text{d}^{2}v_{x}/\text{d}x^{2}]_{x=x_{0}}=0$ as $v_{x}$ is an odd function about $x=x_{0}$ . Equations (3.1) and (3.2) are only necessary conditions for the existence of tearing type eigenfunctions which could either be stable or unstable. Since the right-hand side of (3.2) is always finite, $b_{x}(x_{0})$ and $[\text{d}^{2}b_{x}/\text{d}x^{2}]_{x_{0}}$ cannot be zero, except when both $b_{x}(x_{0})$ and $[\text{d}^{2}b_{x}/\text{d}x^{2}]_{x_{0}}$ approach zero simultaneously.

Equation (3.1) is the mode resonance condition and its solution provides the mode resonance surfaces $x=x_{0}$ . Note that (3.1) is different from the usual resonance condition ( $F=\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}_{0}=0$ ) for the tearing mode. Equation (3.1) has a term due to the electron inertia ( $d_{e}^{2}F^{\prime \prime }$ ) which in EMHD provides an effective dissipation for the growth of tearing instability. It reduces to $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}_{0}=0$ only for equilibrium scale lengths much larger than the electron inertial length (scale length of the effective dissipation). In fact, the mode resonance condition for the MHD tearing mode should also be different from $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}_{0}=0$ and have terms due to the plasma resistivity which provides dissipation for the growth of the MHD tearing instability. This can be seen by applying conditions for the tearing mode eigenfunctions in equation (14) of Furth, Killeen & Rosenbluth (Reference Furth, Killeen and Rosenbluth1963). Assuming uniform density, we obtain for MHD

where $L_{{\it\eta}}={\it\eta}/4{\rm\pi}{\it\omega}$ is the resistive scale length and ${\it\eta}$ is the resistivity. Equation (3.3) reduces to $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}_{0}=0$ only when the equilibrium scale length is much larger than the resistive scale length.

Substituting for the equilibrium magnetic field $\boldsymbol{B}_{0}$ in (3.1), the values of $x_{0}$ can be obtained from following equation

For zero guide field, $B_{z0}=0$ , the only real solution of (3.4) is $x_{0}=0$ , as expected. For finite guide field, $x_{0}\leqslant 0$ , as $x_{0}>0$ is not allowed by (3.4) for $B_{z0}>0$ . The real root of (3.4), $x_{0}/L\leqslant 1$ , is plotted in $L/d_{e}$ – $k_{z}B_{z0}/k_{y}B_{y0}$ parameter space in figure 1. The white space in this plot identifies the regions of $L/d_{e}$ – $k_{z}B_{z0}/k_{y}B_{y0}$ space where no real solution of (3.4) exist. In the limit $L\gg \sqrt{2}d_{e}$ and $k_{z}B_{z0}/k_{y}B_{y0}\gg 1$ , (3.4) reduces to $\tanh ^{3}(x_{0}/L)=k_{z}B_{z0}L^{2}/2k_{y}B_{y0}d_{e}^{2}$ which has no real solution because $\tanh (x_{0}/L)\leqslant 1$ . When $k_{z}B_{z0}/k_{y}B_{y0}<1$ , real solutions of $x_{0}$ exist for any value of $L$ . In this limit, the value of $x_{0}/L$ does not depend on $L$ for $L\gg \sqrt{2}d_{e}$ but does depend on $k_{z}B_{z0}/k_{y}B_{y0}$ . For $L\gg \sqrt{2}d_{e}$ and $k_{z}B_{z0}/k_{y}B_{y0}<1$ , (3.4) can be simplified by neglecting the cubic term to give $x_{0}/L=-\tanh ^{-1}(k_{z}B_{z0}/k_{y}B_{y0})$ , which is independent of $L$ . This result can be derived from the condition $\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}_{0}=0$ which can be obtained from the resonance condition, (3.1), in the limit $L\gg d_{e}$ . We find that the relation, $k_{z}B_{z0}/k_{y}B_{y0}=(1/\sqrt{3})(1+4d_{e}^{2}/3L^{2})$ , approximates very well the boundary (black dashed line in figure 1) between the regions of the existence and non-existence of the mode rational surface for $L<\sqrt{2}d_{e}$ .

Figure 1.

Real root $|x_{0}/L|\leqslant 1$ of (3.4). Colour scales are saturated at $x_{0}/L=-1$ . The black dashed line is the plot of $k_{z}B_{z0}/k_{y}=(1/\sqrt{3})(1+4/3L^{2})$ . No real root exists in the white regions in $L$ – $k_{z}B_{z0}/k_{y}$ space.

The resonance condition, (3.1), is only a necessary condition for the existence of tearing eigenfunctions. Therefore it cannot be used with certainty to predict the existence of tearing instability. However, its violation guarantees that a tearing mode cannot exist. The conditions for the non-existence of the tearing mode, obtained from the non-existence of the resonance surface, can be stated as follows.

3.2. Growth rates

In general, the 3-D unstable modes driven by electron shear flow in EMHD have both an oscillatory and growing nature, i.e. both the real and imaginary parts of the eigenvalue ${\it\omega}$ are finite. However, the 2-D mode with $k_{z}=0$ is purely growing. This follows from (2.2) and (2.3) in the limit $k_{z}=0$ . Adding and integrating (2.2) and (2.3) (for $k_{z}=0$ ) after multiplying ${\it\omega}B_{y0}^{2}v_{x}^{\ast }/d_{e}^{2}{\it\omega}_{ce}^{2}$ (asterisk denotes complex conjugate) to (2.2) and ${\it\omega}b_{x}$ to the complex conjugate of (2.3), we find

The imaginary part of (3.7) gives,

The integrand in square brackets in (3.8) is always positive. Thus (3.8) is satisfied for non-zero ${\it\gamma}$ ( ${\it\omega}_{r}$ ) only if ${\it\omega}_{r}=0$ ( ${\it\gamma}=0$ ). As $k_{y}$ and $L$ vary, the modes change their character from unstable ( ${\it\gamma}\neq 0$ , ${\it\omega}_{r}=0$ ) to stable ( ${\it\gamma}=0,{\it\omega}_{r}\neq 0$ ). Setting ${\it\omega}=0$ in (2.2) and (2.3), a condition for the 2-D tearing instability can be obtined as $0<k_{y}L<1$ . In the case of 3-D perturbations with finite ${\it\omega}_{r}$ and ${\it\gamma}$ , ${\it\omega}_{r}$ is given by a modified whistler dispersion relation which takes into account the electron flow and its gradients (Gaur & Das Reference Gaur and Das2012). In this paper we focus on the growth rates of ESFI to find the dominant electron shear flow driven, tearing or non-tearing, modes as function of $L$ and $B_{z0}$ .

Growth rates in $k_{y}$ – $k_{z}$ space for various values of $L$ and $B_{z0}$ are shown in figure 2. In general, the modes with $kd_{e}\sim 1$ and/or $kd_{e}\gg 1$ have finite growth rates in select regions of $k_{y}$ – $k_{z}$ space. For example, in the panel for $L/d_{e}=0.5$ and $B_{z0}=B_{y0}$ , unstable modes can be seen in the bottom-left ( $kd_{e}\sim 1$ ) and top-right ( $kd_{e}\gg 1$ ) corners of the panel. The eigenfunctions of the select modes chosen from the two corners of the panel are shown in figure 3. The typical signature of tearing mode can be seen in the eigenfunctions for $k_{y}d_{e}=0.8$ and $k_{z}d_{e}=0$ , figure 3. In fact, the 2-D modes with $k_{z}=0$ are always tearing modes and are not affected by the guide field. The eigenfunctions in the other three panels, figure 3(b–d), are different from tearing eigenfunctions. Although the mode with $k_{y}d_{e}=10$ and $k_{z}d_{e}=16$ , figure 3(b), falls in the region (above the white dashed line) of the $k_{y}$ – $k_{z}$ space where a mode rational surface exists, it is not a tearing mode. This is because the existence of mode rational surfaces is only a necessary condition for the tearing mode. The modes in figure 3(c) ( $k_{y}d_{e}=0$ and $k_{z}d_{e}=2.25$ ) and figure 3(d) ( $k_{y}d_{e}=8$ and $k_{z}d_{e}=30$ ) fall in the region of $k_{y}$ – $k_{z}$ space where mode resonance surfaces do not exist. This indicates that tearing modes exist only for very small values of $k_{z}$ when $0<k_{y}L<1$ for given values of $L$ and $B_{z0}$ .

Figure 2.

Growth rates ${\it\gamma}/{\it\omega}_{ce}$ (colour) of ESFI in $k_{y}$ – $k_{z}$ space for $L/d_{e}=0.5$ , 1.0, 2.0 and 5.0 (along column). For each value of $L$ , $B_{z0}/B_{y0}=0$ , 0.5, 1.0 and 5.0. (along row). White dashed lines separate the regions of the existence (on the left or above the line) and non-existence (on the right or below the line) of mode rational surfaces according to the (3.5) and (3.6).

Figure 3.

Real (solid) and imaginary (dashed) parts of eigenfunctions $v_{x}$ (blue) and $b_{x}$ (red) for various values of $k_{y}$ and $k_{z}$ when $L=0.5\,d_{e}$ and $B_{z0}=B_{y0}$ .

One distinctive feature of non-tearing modes is the even parity of $v_{x}$ in contrast to its odd parity for the tearing mode. Also, eigenfunctions become increasingly localized with increasing $k$ . In the limit of large $k$ , the first term on the right-hand side of (2.2) can be neglected (Jain & Büchner Reference Jain and Büchner2014a ). This gives us an equations for  $v_{x}$ .

As $k$ increases, the decay length of the solution of (3.9) decreases and thus the eigenfunctions become increasingly localized. For non-tearing modes, $b_{x}<v_{x}$ which holds stronger as $k$ increases. This can be seen from (2.3) by comparing the left-hand side and right-hand side, giving,

where $k_{x}^{-1}$ is the $x$ -scale length of the eigenfunctions. It is clear from (3.10) that the ratio $b_{x}/v_{x}$ drops as $k$ increases.

The effect of guide field can be understood by noting that the guide field appears in (2.2) and (2.3) through the terms involving $F=k_{z}B_{z0}+k_{y}B_{0}(x)$ and $F-d_{e}^{2}F^{\prime \prime }=k_{z}B_{z0}+k_{y}B_{0}(x)+k_{y}d_{e}^{2}v_{0}^{\prime }(x)$ , where $B_{0}(x)={\hat{y}}\boldsymbol{\cdot }\boldsymbol{B}_{0}$ . For $L\ll d_{e}$ , $d_{e}^{2}v_{0}^{\prime }\sim d_{e}^{2}B_{y0}/L^{2}\gg B_{y0}$ resulting in $F-d_{e}^{2}F^{\prime \prime }\approx k_{y}v_{0}^{\prime }+k_{z}B_{z0}$ . In the opposite limit $L\gg d_{e}$ , $d_{e}^{2}v_{0}^{\prime }\sim d_{e}^{2}B_{y0}/L^{2}\ll B_{y0}$ and $F-d_{e}^{2}F^{\prime \prime }\approx k_{y}B_{0}+k_{z}B_{z0}$ . Thus the conditions under which the guide field affects the growth rate can be stated as follows.

These conditions are similar to the conditions for the non-existence of the tearing mode given by (3.5) and (3.6). Thus, for given values of $L$ and $B_{z0}$ , the guide field affects the modes in the select region of $k_{y}{-}k_{z}$ space given by (3.11) and (3.12). For the modes with $k_{y}/k_{z}\ll B_{z0}/B_{y0}$ (for $L\gg d_{e}$ ) or $k_{y}/k_{z}\ll (L^{2}/d_{e}^{2})(B_{z0}/B_{y0})$ (for $L\ll d_{e}$ ), the guide field term dominates and $F=F-d_{e}^{2}F^{\prime \prime }=k_{z}B_{z0}$ . These modes can be unstable for $L\lesssim d_{e}$ as the term $\propto d_{e}^{2}v_{0}^{\prime \prime }\sim d_{e}^{2}v_{0}/L^{2}\gtrsim v_{0}$ (the only term left causing the instability) survives in the last term on the right-hand side of (2.2). For $L\gg d_{e}$ , these modes are stable as the term $\propto d_{e}^{2}v_{0}^{\prime \prime }\sim d_{e}^{2}v_{0}/L^{2}\ll v_{0}$ can be neglected.

These effects of the guide field can be seen in figure 2. For $L>d_{e}$ , the growth rate is zero in the region where $k_{y}/k_{z}\ll B_{z0}/B_{y0}$ (below white dashed line). As a consequence, the domain of the unstable modes shifts to smaller values of $k_{z}$ as the strength of the guide field increases for a fixed $L$ . For $L<d_{e}$ , e.g.  $L=0.5\,d_{e}$ and $B_{z0}=B_{y0}$ in figure 2, the modes with $k_{y}/k_{z}\lesssim (L^{2}/d_{e}^{2})(B_{z0}/B_{y0})=0.25$ (below and around the white dashed line) have finite growth rates. That is why panels (a3) and (b2) look different from the other panels in figure 2.

Figure 4.

Growth rates ${\it\gamma}_{f}$ (a), wavenumbers $k_{y}^{f}$ (b) and $k_{z}^{f}$ (c) of the fastest growing mode as functions of guide field $B_{z0}$ for various current sheet thicknesses. In (d), ratio ${\it\gamma}_{f}/{\it\gamma}_{f}^{k_{z}=0}$ , where ${\it\gamma}_{f}^{k_{z}=0}$ is the growth rate of the fastest 2-D ( $k_{z}=0$ ) tearing mode as a function of the guide field.

The maximum growth rate drops rapidly with $L$ for a given value of the guide field, but increases slowly with the guide field for fixed current sheet thickness. An electron shear flow instability whose growth rate increases with the strength of the guide field was also reported in 3-D PIC simulations of electron current sheets (Che et al. Reference Che, Drake and Swisdak2011). The variation of the growth rate ( ${\it\gamma}_{f}$ ) of the fastest growing mode (maximum growth rate in $k_{y}$ – $k_{z}$ space) with the strength of the guide field for various values of $L$ is shown in figure 4(a). The rate of increase of the growth rate with the guide field is faster for weak guide field ( $B_{z0}/B_{y0}<1$ ) as compared to the rate for strong guide field ( $B_{z0}/B_{y0}>5$ ). In between the fast and slow rise of the growth rate with the guide field, a plateau forms. The rate of the fast rise of ${\it\gamma}_{f}$ (for weak guide field) increases with $L$ , while the rate of the slow rise (for strong guide field) decreases with $L$ . The range of the guide field values for the plateau depends on  $L$ .

The wavenumbers, $k_{y}^{f}$ and $k_{z}^{f}$ , of the fastest growing mode are shown in figure 4(b,c). The wavenumber $k_{y}^{f}$ makes a jump to large values as soon as the guide field becomes finite, and remains at this value for a range of the values of guide field. On further increasing the guide field, $k_{y}^{f}$ drops and then rises again. This trend of variation of $k_{y}^{f}$ with the guide field can be seen for all values of  $L$ . The jump in the value of $k_{y}^{f}$ is largest for $L=d_{e}$ . The wavenumber $k_{z}^{f}$ has a large jump as soon as the guide field becomes finite only for $L=d_{e}$ . After the initial jump, $k_{z}^{f}$ drops. For $L>d_{e}$ , $k_{z}^{f}$ drops with increasing guide field. For all values of $L$ , $k_{z}^{f}$ saturates at $k_{z}^{f}d_{e}\approx 1$ for large values of guide field. The sudden changes in ${\it\gamma}_{f}$ , $k_{y}^{f}$ and $k_{z}^{f}$ as soon as the guide field becomes finite indicate that the inclusion of the guide field triggers a new kind of unstable mode, different from the tearing mode.

The eigenfunctions $v_{x}$ and $b_{x}$ obtained from (2.2) and (2.3) are, in general, complex and contain phase and magnitude information in their real and imaginary parts. The physical electron velocity ( $V_{x}$ ) and magnetic field ( $B_{x}$ ) perturbations are related to the complex eigenfunctions, $v_{x}(x)=|v_{x}|\exp (\text{i}{\it\theta}_{v})$ and $b_{x}(x)=|b_{x}|\exp (\text{i}{\it\theta}_{b})$ , as $V_{x}=\text{Re}[v_{x}(x)\text{e}^{\text{i}(k_{y}y+k_{z}z-{\it\omega}t)}]$ and $B_{x}=\text{Re}[b_{x}(x)\text{e}^{\text{i}(k_{y}y+k_{z}z-{\it\omega}t)}]$ . Here $|v_{x}|^{2}=[\text{Re}(v_{x})]^{2}+[\text{Im}(v_{x})]^{2}$ , $|b_{x}|^{2}=[\text{Re}(b_{x})]^{2}+[\text{Im}(b_{x})]^{2}$ , $\tan ({\it\theta}_{v})=\text{Im}(v_{x})/\text{Re}(v_{x})$ and $\tan ({\it\theta}_{b})=\text{Im}(b_{x})/\text{Re}(b_{x})$ . Substituting for $v_{x}$ and $b_{x}$ relates the physical variables $V_{x}$ and $B_{x}$ to the absolute values $|v_{x}|$ and $|b_{x}|$ which, for the fastest mode, are plotted in figure 5 in order to distinguish between tearing and non-tearing modes.

Figure 5.

Absolute values of complex eigenfunctions $v_{x}$ (blue) and $b_{x}$ (red) corresponding to the fastest growing mode for $L/d_{e}=0.5,1,1.5,2$ and 5 (varying along a column), and $B_{z0}/B_{y0}=0,1,3,5$ and 10 (varying along a row). Horizontal axis is in units of $L$ and shows only a small region in $x$ .

For the tearing mode, the $x$ -component of the physical magnetic field ( $B_{x}$ ) and velocity ( $V_{x}$ ) should be finite and zero, respectively, at the mode rational surface. In figure 5, panels for $B_{z0}=0$ , $L/d_{e}=1$ ( f) and $B_{z0}=0,L/d_{e}=1.5$ (k) show the typical signature of the tearing mode, namely, $|v_{x}|=0$ and $|b_{x}|\neq 0$ (therefore also for $V_{x}$ and $B_{x}$ ) at the same location $x=0$ (mode resonance surface). No other panel shows the simultaneous occurrence of $|v_{x}|=0$ and $|b_{x}|\neq 0$ at a mode resonance surface. Therefore, the tearing mode is the fastest growing mode in a very small range (around $L=d_{e}$ and $B_{z0}=0$ ) of $L{-}B_{z0}$ parameter space.

Although the fastest growing mode is non-tearing in the presence of a finite guide field, the 2-D tearing mode ( $k_{z}=0$ ) is not affected by the guide field because the guide field disappears from (2.2) and (2.3) for $k_{z}=0$ . For $L=d_{e}$ and $B_{z0}=B_{y0}$ , the growth rate of the fastest 2-D tearing mode ( $k_{y}d_{e}=0.4$ ) is comparable to that of the fastest growing 3-D non-tearing mode (figure 2). In order to determine the relative importance of the tearing and non-tearing modes as a function of $L$ and $B_{z0}$ , we compare the growth rate of the fastest 3-D mode ( ${\it\gamma}_{f}$ ) and that of the fastest 2-D ( $k_{z}=0$ ) tearing mode ( ${\it\gamma}_{f}^{k_{z}=0}$ ) in figure 4(d). The ratio ${\it\gamma}_{f}/{\it\gamma}_{f}^{k_{z}=0}$ increases with both the current sheet thickness and the strength of the guide field. For zero guide field, the ratio ${\it\gamma}_{f}/{\it\gamma}_{f}^{k_{z}=0}$ is of the order of unity (varies from 1 to 3 for $L/d_{e}=1$ to 6). The ratio increases with the strength of the guide field but remains of order unity for thinner current sheets ( $L/d_{e}=1$ and 2) for guide fields as large as $B_{z0}/B_{y0}=10$ . The initial rate of increase of the ratio with guide field is larger for thicker current sheets. Therefore the ratio increases to much larger values even for $L>2\,d_{e}$ . It is expected that the tearing mode can grow simultaneously with the non-tearing modes for thin current sheets and small values of guide field.