Nonlinear oscillations of coalescing magnetic flux ropes

An analytical model of highly nonlinear oscillations occurring during a coalescence of two magnetic flux ropes, based upon two-fluid hydrodynamics, is developed. The model accounts for the effect of electric charge separation, and describes perpendicular oscillations of the current sheet formed by the coalescence. The oscillation period is determined by the current sheet thickness, the plasma parameter $\beta$, and the oscillation amplitude. The oscillation periods are typically greater or about the ion plasma oscillation period. In the nonlinear regime, the oscillations of the ion and electron concentrations have a shape of a narrow symmetric spikes.

Figure 1

Formation of a thin 1.5D current sheet (the hatched vertical slab) by coalescence instability. $j_{\mathrm{z}}, z$ component of the plasma current generating the poloidal magnetic field; $\lambda$, scale length of the poloidal magnetic field $B_{\mathrm{y}}$ where the field lines can be considered as straight. $\lambda$ is small in comparison with the radii of the colliding magnetic ropes $R$.

Figure 2

(Top) Bernoulli pseudopotential $U_{\mathrm{B}}\left(A\right)$ (14) plotted for $\gamma =3$ and for $\overline{\varphi }=0.685$ (left solid), $\overline{\varphi }=0.3$ (left dashed), and $\overline{\varphi }=5$ (right solid), $\overline{\varphi }=2$ (right dashed). The upper horizontal dotted line in the left panel shows the energy level above which $U_{\mathrm{B}}\left(A\right)$ experiences the change of behavior; the bottom dashed line shows the energy level of the oscillation shown in Fig. 3, left bottom panel. The horizontal dotted line in the right panel indicates the energy level of the nonlinear signal shown in Fig. 3, right top and bottom panels. (Bottom) The oscillation period-amplitude dependence shown for $\overline{\varphi }=2\times {10}^{-4}\left(\mathrm{a}\right),{10}^{-2}\left(\mathrm{b}\right),\mathrm{and}0.3\left(\mathrm{c}\right)$ (the small $\overline{\varphi }$ regime, left panel), and for $\overline{\varphi }=5\left(\mathrm{a}\right),10\left(\mathrm{b}\right),100\left(\mathrm{c}\right)$ (the large $\overline{\varphi }$ regime, right panel), and $\overline{\varphi }\to \infty$ (the period equals to $2\pi$, dashed line). The period is measured in units of ${(V_{\mathrm{s}}/V_{\mathrm{A}})}^{1/2}{\omega }_{\mathrm{i}}^{-1}$.

Figure 3

(Top) Variations of the electron $n_{\mathrm{e}}=n_0/a$ (thick lines) and ion $n_{\mathrm{i}}=n_0/b$ (thin lines) concentrations, where the scale factors $a$ and $b$ are obtained from the numerically obtained solutions $A\left(\tau \right)$ of Eq. (9) with $\gamma =3, \overline{\varphi }=5, A\left(0\right)=1$, and ${\left.\frac{dA}{d\tau }\right|}_{\tau =0}=0.04$ (left panel, quasilinear oscillation), and ${\left.\frac{dA}{d\tau }\right|}_{\tau =0}=0.155$ (right panel, nonlinear oscillation). (Bottom left) Variations of $n_{\mathrm{e}}$ (dotted line) and $n_{\mathrm{i}}$ (solid line, almost indistinguishable from the dotted line) for $\gamma =3, \overline{\varphi }=0.3, A\left(0\right)=1$, and ${\left.\frac{dA}{d\tau }\right|}_{\tau =0}=0.426$ (highly nonlinear case). Both functions $n_{\mathrm{e}}$ and $n_{\mathrm{i}}$ are normalized to $n_0{\overline{a}}^{-1}$. (Bottom right) Variation of the electrostatic field energy $E_x^2$ normalized to ${\left(4\pi e{n}_0\lambda {\overline{a}}^{-1}\right)}^2$, generated by the local charge separation, in the quasilinear (thick line) and nonlinear (thin line) regimes of the current sheet oscillation, shown in the top panels. The vertical axes in all panels are plotted on a logarithmic scale. The time $\tau$ is measured in units of ${(V_{\mathrm{s}}/V_{\mathrm{A}})}^{1/2}{\omega }_{\mathrm{i}}^{-1}$.