Transient growth in stochastic Burgers flows

Figure 1.  (a) Space-time evolution of the solution $u(t,x)$ and (b) history of the enstrophy ${{\mathcal{E}}(u(t))}$ in a solution of the deterministic Burgers equation with an extreme initial condition $\tilde{g}_{\mathcal{E}_0,T}$. In figure (a) the level sets of $u(t,x)$ are plotted with increments of 0.1.

Figure 2.  Errors in the numerical approximations of $\mathcal{E}(\mathbb{E}[u(T)])$ (blue lines and circles) and $\mathbb{E}[\mathcal{E}(u(T))]$ (green lines and squares) as functions of (a) the spatial discretization parameter $M$ with $N = 20,000$ and $S = 1000$ fixed, (b) the temporal discretization parameter $N$ with $M = 1024$ and $S = 1000$ fixed and (c) the sampling discretization parameter $S$ with $M = 1024$ and $N = 20,000$ fixed. The initial data used was $\tilde{g}_{\mathcal{E}_0,T}$ with $\mathcal{E}_0 = 10$ and $T = 1$, and the errors are evaluated with respect to the reference solutions computed with $M = 1024$, $N = 20,000$ and $S = 1000$. The dashed black lines correspond to the power laws (a) $CM^{-2}$, $CM^{-3}$, and $CM^{-4}$, (b) $CN^{-1}$, and (c) $CS^{-1/2}$ with suitably adjusted constants $C$.

Figure 3.  Optimal initial conditions $\tilde{g}_{\mathcal{E}_0,T}(x)$ for $\mathcal{E}_0 = 10$ and $T$ ranging from $10^{-3}$ to $1$ [5] (arrows indicate the directions of increase of $T$).

Figure 4.  Dependence of (a) the enstrophy $\mathcal{E}(T)$ at a final time $T$ and (b) the maximum enstrophy $\max_{t \in [0,T]} \mathcal{E}(t)$ on the initial enstrophy $\mathcal{E}_0$ for the optimal initial data $\tilde{g}_{\mathcal{E}_0,T}$ with $T$ in the range from $10^{-3}$ to $1$. Arrows indicate the direction of increasing $T$ and the dashed lines correspond to the power law $C \, \mathcal{E}_0^{3/2}$.

Figure 5.  [Small noise case: $\sigma^2 = 10^{-2}$] (a) Sample stochastic solution ${ u(t,x)}$ as a function of space and time (the level sets are plotted with the increments of $0.1$), (b) evolution of enstrophy of two sample stochastic solutions $\mathcal{E}({ u(t;\omega_s)})$, $s = 1,2$, (green dash-dotted lines), the enstrophy of the deterministic solution $\mathcal{E}(t)$ (black solid line), the expected value of the enstrophy $\mathbb{E}[\mathcal{E}({ u(t)})]$ (blue dashed line) and the enstrophy of the expected value of the solution $\mathcal{E}(\mathbb{E}[{ u(t)}])$ (red dotted line). The initial data used was $\tilde{g}_{\mathcal{E}_0,T}$ with $\mathcal{E}_0 = 10$ and $T = 1$. The inset in figure (b) shows details of the evolution during the subinterval $[0.35,0.65]$.

Figure 6.  [Large noise case: $\sigma^2 = 1$] (see previous figure for details).

Figure 7.  The expected value of the enstrophy $\mathbb{E}[\mathcal{E}({ u(t)})]$ (dashed lines), the enstrophy of the expected value of the solution $\mathcal{E}(\mathbb{E}[{ u(t)}])$ (dotted lines) and the enstrophy $\mathcal{E}(t)$ of the deterministic solution (thick solid line) as functions of time for the initial condition $\tilde{g}_{\mathcal{E}_0,T}$ with $\mathcal{E}_0 = 10$, $T = 1$ and different noise levels $\sigma^2$ in the range from $10^{-2}$ to $1$ (the direction of increase of $\sigma^2$ is indicated by arrows).

Figure 8.  Normalized PDFs of the maximum enstrophy values $\max_{t \ge 0} {\mathcal{E}}(u(t,\omega))$ for the cases with the initial condition $\tilde{g}_{\mathcal{E}_0,T}$ with $T = 1$ and (a) $\mathcal{E}_0 = 10$, (b) $\mathcal{E}_0 = 10^3$. The noise levels $\sigma^2$ are equal to $10^{-2}$ (green lines and crosses), $10^{-1}$ (blue lines and squares) and $1$ (red lines and circles). To obtain these plots, $S = 10^5$ samples were collected in each case and sorted into $30$ equispaced bins. The solid lines correspond to the standard Gaussian distributions.

Figure 9.  (a) The values at $T = 1$ and (b) the maximum values attained in $[0,T]$ of the expected value of the enstrophy $\mathbb{E}[\mathcal{E}({ u(t)})]$ (dashed lines), the enstrophy of the expected value of the solution $\mathcal{E}(\mathbb{E}[{ u(t)}])$ (dotted lines) and the enstrophy $\mathcal{E}(t)$ of the deterministic solution (thick solid line) as functions of the initial enstrophy $\mathcal{E}_0$ for the initial condition $\tilde{g}_{\mathcal{E}_0,T}$ with $\mathcal{E}_0 = 10$, $T = 1$ and different noise levels $\sigma^2$ in the range from $10^{-2}$ to $1$ (the direction of increase of $\sigma^2$ is indicated by arrows)

Figure 10.  [Small noise case: $\sigma^2 = 10^{-2}$] Dependence of (a) the enstrophy of the expected value of the solution $\mathcal{E}(\mathbb{E}[{ u(T)}])$ and (b) the expected value of the enstrophy $\mathbb{E}[\mathcal{E}({ u(T)})]$ on the initial enstrophy $\mathcal{E}_0$ using the initial condition $\tilde{g}_{\mathcal{E}_0,T}$ with $T$ varying from $10^{-3}$ to $1$. In (a) the values of $T$ are marked near the right edge of the plot, whereas in (b) the direction of increasing $T$ is indicated with an arrow. The dashed lines correspond to the power law $C \, \mathcal{E}_0^{3/2}$.

Figure 11.  [Large noise case: $\sigma^2 = 1$] (see previous figure for details).

Figure 12.  Dependence of (a) the maximum enstrophy of the expected value of the solution $\max_{t\in[0,T]} \mathcal{E}(\mathbb{E}[{ u(t)}])$ and (b) the maximum expected value of the enstrophy $\max_{t\in[0,T]} \mathbb{E}[\mathcal{E}({ u(t)})]$ on the initial enstrophy $\mathcal{E}_0$ using the initial conditions $\tilde{g}_{\mathcal{E}_0,T}$ and with noise magnitudes proportional to $\mathcal{E}_0$, cf. (14), with $C_{\sigma}$ in the range from $10^{-3}$ to $10^{-1}$ (arrow indicate the direction of increase of $C_{\sigma}$). The parameter $T$ is chosen to maximize $\max_{t\in[0,T]} \mathcal{E}(\mathbb{E}[{ u(t)}])$ in (a) and $\max_{t\in[0,T]} \mathbb{E}[\mathcal{E}({ u(t)})]$ in (b). The thick black solid line corresponds to the quantity $\max_{t \in [0,T]} \mathcal{E}(t)$ obtained in the deterministic case, whereas the thin black solid line in (a) represents the power law $\mathcal{E}_0^1$.