Large-Scale Pattern Formation in the Presence of Small-Scale Random Advection

Despite the presence of strong fluctuations, many turbulent systems such as Rayleigh-Bénard convection and Taylor-Couette flow display self-organized large-scale flow patterns. How do small-scale turbulent fluctuations impact the emergence and stability of such large-scale flow patterns? Here, we approach this question conceptually by investigating a class of pattern forming systems in the presence of random advection by a Kraichnan-Kazantsev velocity field. Combining tools from pattern formation with statistical theory and simulations, we show that random advection shifts the onset and the wave number of emergent patterns. As a simple model for pattern formation in convection, the effects are demonstrated with a generalized Swift-Hohenberg equation including random advection. We also discuss the implications of our results for the large-scale flow of turbulent Rayleigh-Bénard convection.

Figure 1

Effect of random advection on type-I (left) and type-II (right) instabilities. As the random advection amplitude $Q$ is increased (from light to dark blue) for a fixed $r$, both the maximum growth rate and the corresponding wave number of the linear dispersion relation are shifted. Here, $k_c$ denotes the maximum-growth wave number of the case without advection ($Q=0$). The dashed line shows the maximum growth rate as a function of $k$ and $Q$.

Figure 2

Formation of a stripe pattern in the Swift-Hohenberg equation with random advection. Single realization (top left), ensemble-averaged field (top right), mean squared fluctuations (bottom left) and mean cubed fluctuations (bottom right) from simulations of Eq. (5) in the statistically stationary state. Parameters are $\lambda ={\lambda }_c/10$, $Q=0.1$, and $r=0.2$. The simulations have been conducted on a periodic domain with $L=20\pi$ on a $256\times 256$ grid. A comparison of the field amplitudes shows that the fluctuations are negligible compared to the averaged field.

Figure 3

(a) Root mean squared amplitude in the stationary state as a function of control parameter $r$ for varying amplitude $Q$ of the random advection field. All other simulation parameters are chosen as in Fig. 2. Compared to the bifurcation curve without random advection (purple line), the onset is shifted. The theoretically predicted shift of onset (indicated with vertical lines) agrees well with the numerical results. (b) For the renormalized control parameter $r^{*}$, all cases collapse on the theoretically predicted master curve.

Figure 4

Numerically obtained dominant wave number of the individual realizations (blue squares, simulations with $L=40\pi$ on a $512\times 512$ grid with $\lambda ={\lambda }_c/10$ and $r=0.9$) shown along with the theoretically predicted shifted critical wave number $k_c^{*}$ (dashed black line). As an illustration, the insets show individual realizations (scaled to unit maximum amplitude) for different $Q$.