Mixing-Induced Global Modes in Open Active Flow

We describe how local mixing transforms a convectively unstable active field in an open flow into absolutely unstable. Presenting the mixing region as one with a locally enhanced effective diffusion allows us to find the linear transition point to an unstable global mode analytically. We derive the critical exponent that characterizes weakly nonlinear regimes beyond the instability threshold and compare it with numerical simulations of a full two-dimensional flow problem.

Figure 1

(a) Quasi-one-dimensional open flow with a localized mixing zone. Panels (b) and (c) illustrate the construction of the nonlinear global mode, they show qualitative profiles $u(x)$ and $z(x)=u^{-1}\frac{du}{dx}$ in the linear approximation at the criticality (dashed line) and in nonlinear regime slightly beyond criticality (full line). In the latter case the profile is nearly linear for $x<x_r$ but deviates due to nonlinear terms for $x>x_r$, see discussion of Eq. (14). Regions source, tail, and plateau are explained in the text.

Figure 2

Lyapunov exponent characterizing stability of the global mode mixed by periodically blinking vortex (10). A similar picture holds for a stationary vortex.

Figure 3

The critical exponent $\beta$ calculated for the model (1) (symbols), compared with theoretical prediction (16) (lines). One can clearly see the crossover between two regimes of the field saturation in dependence on parameter $a$, the latter is related to $v$ in (16) via $v\sim a^{-1/2}$.

Figure 4

The active field behind a vortex with $W=4$ placed at $x=y=0$, for $V=1$, $a=0.5$, $D_0=0.3$, $p=2$.