Emergence of dissipative structures in current-carrying superconducting wires

We discuss the emergence of a spontaneous temperature and critical current spatial modulation in current-carrying high-temperature superconducting wire. The modulation of the critical current along the wire on a scale of 3–10 mm forces a fraction of the transport current to crisscross the resistive interface between the superconducting film and normal metal stabilizer attached to it. This generates additional heat that allows such a structure to be self-sustainable. Stability and the conditions for experimental observation of this phenomenon are also discussed.

Figure 1

(Color online) A sketch of the cross section of coated conductor (not to scale). A thin superconducting film is sandwiched between the copper stabilizer (1) and metal substrate (2). The resistive interface between copper and YBCO and the insulating buffer between YBCO and substrate are not shown.

Figure 2

(Color online) The solid line is a sketch of the temperature-dependent critical current density $J_c$. Temperature increases from left to right. The dashed line $J_s$ is the current density that flows through the superconductor. The dotted line is the constant total current density. The point $x_c$ demarcates the boundary between the subcritical $E_s=0$ and critical $E_s>0$ sections.

Figure 3

(Color online) Sketch of the right-hand sides of Eqs. (36, 37, 39) as a function of temperature in the limit of uniform temperature. Solid lines is the heat source; dashed lines indicate cooling term for two values of cooling constant ${\kappa }_0$.

Figure 4

(Color online) Sequence of temperature profiles as they develop from the initial Gaussian perturbation visible in the foreground. All profiles correspond to the interface resistance $r=1.4$, Eq. (38). The width of the initial temperature profile $\delta =1.4$, Eq. (43). The range of the plots are $0<\gamma t\le 50$ and $-40\le x/l_T\le 40$. (a) ${\kappa }_0=0.1$; conventional normal zone, but it propagates with greater speed than that at $r⪡1$. (b) ${\kappa }_0=0.35$; conventional normal zone and temperature ripples emerge simultaneously, but propagate with different speed. (c) Cryostable condition $\left({\kappa }_0=0.55>{\kappa }_c\right)$. Only the temperature ripples are triggered.

Figure 5

(Color online) Profile of the $J_c$ ripples ($r=1.4$, ${\kappa }_0=0.4$) at a certain moment of time, $\gamma t=25$. The inset shows a magnified view of the same profile. The periodicity of the $J_c$ ripple is $\approx 2.7l_T$. The peak temperature is ${\theta }_p\approx 0.24–0.26$ in agreement with the exact value $1-r^{-1}$, Eq. (51).

Figure 6

(Color online) The initial condition gives rise to two propagating fronts that eventually merge. Here $r=1.5$, ${\kappa }_0=0.4$, and ${\theta }_0=-1$. There is a domain boundary between the two merged fronts with slightly irregular temperature variation.

Figure 7

(Color online) Tristable operation which corresponds to ${\kappa }_0=0.4$, $r=1.4$, and ${\theta }_0=-1$. The initial profile evolves into $J_c$ ripples. A heater, Eq. (54), is switched on at $\gamma t=20$ and switched off at $\gamma t=25$. It triggers the transition to the normal state. The evolution of the system depicted in the figure corresponds to $b=1.5$, ${\delta }_1=1.4$.

Figure 8

(Color online) Shown here is view “from above.” The transition to $J_c$-ripples mode is triggered by the initial Gaussian profile. The negative power source collapses temperature in its vicinity, but does not destroy the ripple mode. Parameters here are the same as in Fig. 6.

Figure 9

(Color online) View “from above.” (a) A stationary bimodal solution that evolves from the initial Gaussian perturbation. (b) A more complex profile emerging under the action of the heater, Eq. (54). In both cases $r=1.2$, ${\kappa }_0=0.5$, and ${\theta }_0=-1$.

Figure 10

(Color online) Shown are long lived perturbations evolving from the initial Gaussian profile with $\delta =1.4$. (a) The value of $r=1.0426$ is very close to threshold $r=1$ above which the effective diffusivity can be negative. The profile lingers for a long time, but eventually disappears. (b) At slightly greater $r=1.0427$ the normal zone emerges and starts to propagate. In both cases ${\kappa }_0=0.35$, ${\theta }_0=-1$, and $\delta =1.4$.

Figure 11

(Color online) (a) The ripple mode is triggered by the initial Gaussian profile. The pulsations of temperature with frequency $2\omega =1.5\gamma$ are caused by the heat source given by Eq. (55). Here $r=1.5$, ${\kappa }_0=0.5$, ${\theta }_0=-1$, $\delta ={\delta }_1=1.4$, and $b=1$. (b) The initial Gaussian profile gives rise to a bimodal stationary soliton. The temperature pulsations are confined within the walls of the soliton. Here $r=1.2$, ${\kappa }_0=0.55$, ${\theta }_0=-1$, $\delta ={\delta }_1=1.4$, and $b=0.3$.

Figure 12

The range of parameters $\left\{r,{\kappa }_0\right\}$ specific for different regimes. The value of ${\theta }_0=-1$ and the width of the initial perturbation $\delta =1.4$ are the same for all combinations of $\left\{r,{\kappa }_0\right\}$. Area (1) corresponds to conventional NZP. In area (2) $T$ ripples and NZP coexist. In area (3) $T$ ripples are the second mode of operation. Area (4) corresponds to stationary solitons, and area (5) is the range of cryostability.