Vortex motions in coupled phase oscillator lattices with inertia under shear stress

We propose a coupled phase oscillator model with inertia and study the vortex motion in the model when the external force is applied at the boundaries. The vortex exhibits a glide motion when the external force is larger than a critical value. We find a transition from the pair annihilation to passing for the collision of the vortex and antivortex when the external force is changed. In the parameter range of the passing, a single vortex exhibits a reciprocal motion, which leads to desynchronization. When the external force is further increased, the multiplication of vortices occurs and the jump of the frequency profile increases. The desynchronization induced by the vortex motion is analogous to the plastic flow induced by the dislocation motion under the shear stress in solids. In perfect crystals without dislocations, the plastic flow hardly occurs. We further show that a vortex ring is generated when a vortex line passes through an impurity region, which corresponds to the Orowan loop in the theory of plasticity.

Figure 1

(a) Stationary profile of ${\varphi }_{i,j}$ at $i=L_x/2$ at $K=1$, $F=0.5$, and $d=0.2$. (b) ${\varphi }_{i,j}$ at $i=L_x/2$, at $F=0.95$, and 1.05 for $K=1$ and $d=0.2$.

Figure 2

Phase configuration of a vortex.

Figure 3

(a) Time evolution of the $x$ coordinate of the dislocation position at $F=0.1018$ and 0.15 for $K=1$ and $d=0.2$. (b) Relationship between $F$ and the stick time $ST$ for $K=1$ and $d=0.2$. (c) Time evolution of the $x$ coordinate of the vortex position at $F=0.1018$ for $D=7.3\times {10}^{-5}$ and 0.094 for $K=1$.

Figure 4

(a) Average velocity as a function of $F$ at $K=1,2$ and 4 for $d=0.2$. (b) Critical value $F_c$ as a function of $K$ for $d=0.2$.

Figure 5

(a) Time evolutions of the position of the vortex at $F=0.03$, 0.04, 0.05, 0.1, and 0.2 for the conservative system of $d=0$ and $K=1$. (b) Time evolution of $\varphi (i,j)$ at $j=L_y/2$ at $F=0.3$, $K=1$, and $d=0$.

Figure 6

(a) Time evolutions of $\varphi (i,j)$ at $j=L_y/2$ at $F=0.25$ for $K=1$ and $d=0.05$. (b) Snapshot profile of $(cos{\varphi }_{i,j},sin{\varphi }_{i,j})$ at $t=220$ before the pair annihilation. (c) Snapshot profile of $(cos{\varphi }_{i,j},sin{\varphi }_{i,j})$ at $t=240$.

Figure 7

(a) Time evolutions of $\varphi (i,j)$ at $j=L_y/2$ at $F=0.3$ for $K=1$ and $d=0.05$. (b) Snapshot profile of $(cos{\varphi }_{i,j},sin{\varphi }_{i,j})$ at $t=175$ before the collision. (c) Snapshot profile of $(cos{\varphi }_{i,j},sin{\varphi }_{i,j})$ at $t=180$.

Figure 8

(a) Time evolutions of $\varphi (i,j)$ at $j=L_y/2$ at $F=0.27$ for $K=1$ and $d=0.05$. (b) Stationary profile of $(cos{\varphi }_{i,j},sin{\varphi }_{i,j})$ at $F=0.27$. (c) Critical line below which the pair annihilation occurs in a parameter space $(d,F)K=1$.

Figure 9

(a) Time evolution of the $x$ coordinate of the vortex at $d=0.01$ and $F=0.4$. (b) Time evolution of ${\varphi }_{i,j}$ for $i=300$ (solid line) and 100 (dashed line) at $j=74$. (c) Frequency ${\omega }_j$ as a function of $j-L_y/2$ at $F=0.4$ and $d=0.01$.

Figure 10

(a) Time evolution of the $x$ coordinate of the vortex at $d=0.01$ and $F=0.5$. (b) Time evolution of the $y$ coordinate of the vortex at $d=0.01$ and $F=0.4$. (c) Frequency ${\omega }_j$ as a function of $j$ at $F=0.5$ and $d=0.01$. (d) Relationship between the average number of vortices and the average frequency $\overline{\omega }$ obtained by changing $F$ at $K=1$ and $d=0.01$. The dashed line is $\overline{\omega }=0.035N$.

Figure 11

Five snapshot patterns of the vortex ring at $t=0$, 10, 20, 30, and 40 in the section of $j=L_y/2$ for $d=0.3$ and $F=0.3$.

Figure 12

Numerical results of the three-dimensional model at $d=0.03$ and $F=0.43$. (a) Time evolution of the total number $S$ of elemental square loops with nonzero vorticity. (b) Snapshot of vortex lines at $t=525$. (c) Snapshot of vortex lines at $t=530$.

Figure 13

Numerical results of the three-dimensional model at $d=0.012$ and $F=0.5$. (a) Snapshot of vortex lines at $t=750$. (b) Time evolution of $S$. (c) Frequency ${\omega }_j$ as a function of $j$. (d) Relationship between $S$ and the average frequency $\overline{\omega }$ for $55<j<80$ obtained by changing $F$ for $0.35\le F\le 0.5$ at $K=1$ and $d=0.012$.

Figure 14

(a) Time evolution of the vortex number at $F=0.75$ and $d=0.4$. (b) Frequency of the oscillation of vortex number as a function of $F$ at $d=0.4$. (c) Time evolution of the vortex number at $F=0.75$ and $d=0.05$.

Figure 15

(a) Time evolution of $S$ of the vortex ring at $d=0.5$ and $F=0.9$. (b) Three snapshots of the vortex ring at $t=2190$, 2220, and 2250. (c) Frequency $\omega$ of the oscillation of $S$ as a function of $F$ for $d=0.5$.

Figure 16

(a) Time evolution of ${\varphi }_{i,j,k}$ at the section of $j=L/2$ and $k=L/2$ for $L=100$ at $d=0.1$ and $F=0.2$, and $L=100$. The vortex line is initially set at $i=L/4$ and $j=L/2$ and an impurity region of radius 3 is set at the center $i=L/2$, $j=L/2$, and $k=L/2$. (b) Time evolution of the length $S$ of the vortex ring at $d=0.1$ and $F=0.2$, and $L=100$. (c) Three snapshots of the vortex lines at $t=175,300$, and 400.