Figure 1

Projections of a scroll ring in the BZ reaction under an applied electrical currrent. Top row: top projection. Bottom row: lateral projection. The applied current density $J=40\mathrm{mA}{\mathrm{cm}}^{-2}$ pointed to the right. (a) At the beginning, the unit vector $\mathbf{S}$ was perpendicular to $\mathbf{J}$. The projections of the scroll ring appeared as circles (top view) and two counterrotating spirals (lateral view). (b) The applied current caused a continuous reorientation of the scroll ring, until (c) $\mathbf{S}$ was antiparallel to $\mathbf{J}$. The dimensions of the images are $17\times 8.8{\mathrm{mm}}^2$.Reuse & Permissions

Figure 2

(Color online) Dynamics of the scroll ring in the BZ reaction under an applied electrical current density $J=40\mathrm{mA}{\mathrm{cm}}^{-2}$. Circles show core centers of two counterrotating spirals in the lateral projection at $t=0–76\min$. The radius $r$ (i.e., $r=\mid \mathbf{r}\mid$) and center $(x,z)$ of the scroll ring were estimated as half of the distance and the center of mass of a pair of the core centers, respectively. The unit vector $\mathbf{S}$ is perpendicular to $\mathbf{r}$ and accounts for the spiral rotating direction. The orientation $\theta$ is the angle between $\mathbf{S}$ and $\mathbf{J}$.Reuse & Permissions

Figure 3

(Color online) Reorientation of the scroll ring under an electrical current density $J=40\mathrm{mA}{\mathrm{cm}}^{-2}$. Top and bottom graphs show the dynamics of the orientation $\left(\theta \right)$ and the radius $\left(r\right)$ of the scroll ring, respectively. Circles and squares indicate results from experiments with and without applied current, respectively. During the reorientation under $J=40\mathrm{mA}{\mathrm{cm}}^{-2}$, the angle $\theta$ increased from 90° to 180° and the radius $r$ decreased, however, slower than in the case of $J=0$. Solid and dotted lines show numerical calculations using the theory of local filament dynamics under parameter gradients with $J=32\mathrm{mA}{\mathrm{cm}}^{-2}$ and $J=0$, respectively.Reuse & Permissions

Figure 4

Manipulation of a scroll ring by a stepwise variation of the electrical current. Circles show the position of core centers of two counterrotating spirals in the lateral projection at $200\text{−}\mathrm{s}$ intervals. The current density $\mathbf{J}$ was antiparallel to the unit vector $\mathbf{S}$ of the scroll ring ($\mathbf{J}$ pointed to the left), causing the scroll ring to drift to the right without reorientation. For $0<t<37\min$ (a to b), $J$ was kept constant at $40\mathrm{mA}{\mathrm{cm}}^{-2}$ and the scroll ring drifted to the right with decreasing radius. At $t=37\min$ (b), $J$ was increased to $53\mathrm{mA}{\mathrm{cm}}^{-2}$; however, the radius of the scroll ring still contracted. Shortly before self-annihilatition occurred, $J$ was increased to $106\mathrm{mA}{\mathrm{cm}}^{-2}$ ($t=54\min$, at c). Under this strong applied field, the scroll ring expanded. At $t=67\min$ (d), the external field was switched off and the scroll ring contracted again at an approximately fixed position.Reuse & Permissions

Figure 5

The $xz$ projection of scroll ring filaments in the Oregonator model under an applied electric field $E=0.34$ at $t=0$, 9.2, 18.4, 27.6, 32.9, 38.9, and 42.3 $(a–g)$. Their shape and orientation changed with time. During the reorientation, the pattern of filament projection changed from an almost straight (a) to wavy (b), and curved line (c, d). Towards the end of the simulations (e–g), the projection pattern returned to a straight line before the scroll ring self-annihilated.Reuse & Permissions