Dynamics of partially thermalized solutions of the Burgers equation

The spectrally truncated, or finite dimensional, versions of several equations of inviscid flows display transient solutions which match their viscous counterparts, but which eventually lead to thermalized states in which energy is in equipartition between all modes. Recent advances in the study of the Burgers equation show that the thermalization process is triggered after the formation of sharp localized structures within the flow called “tygers.” We show that the process of thermalization first takes place in well defined subdomains, before engulfing the whole space. Using spatio-temporal analysis on data from numerical simulations, we study propagation of tygers and find that they move at a well defined mean speed that can be obtained from energy conservation arguments.

Figure 1

Evolution of $u(x,t)$ at different times, from (a) $t=1$ to (f) $t=10$, in a simulation with $k_G=5461$ and a single-mode initial condition given by Eq. (6). The shock is formed at $t=1$, and the tyger has developed and started to collapse at $t=1.1$. Afterwards each front of the tyger advances, swallowing up the rest of the solution. (a) $t=1.0$, (b) $t=1.1$, (c) $t=3.0$, (d) $t=4.1$, (e) $t=7.0$, and (f) $t=10.0$.

Figure 2

Spatiotemporal evolution of $u(x,t)$: (a) Evolution in real space as a function of space and time, and (b) evolution in Fourier space as a function of frequency and wave number. The spreading of the tyger after $t=1.1$ is well described by a velocity (marked in both plots with a green solid line) equal to $2/\pi$. This is the mean velocity of propagation of each thermalized subdomain formed to the right and left of the tyger when the tyger appears.

Figure 3

Probability density function (PDF) of $u(x,t)$ at different times as (blue) solid lines; each (green) dashed line is the proposed PDF for that instant. At the time of shock formation (a) the typical PDF of a trigonometric function can be observed, then the tyger appears at (b). At intermediate times the solution is partially thermalized and a bimodal Gaussian distribution gives a good approximation to the data, as seen in (c) and (d); the mean and standard deviation of each thermalized mode are such that the tyger front matches the statistical properties of the solution in the subdomain where it lives. At later times, (e) and (f), after the two fronts meet, the system reaches the last stages of thermalization and the solution now matches the PDF of white noise with statistical properties close to those of the thermal equilibrium. (a) $t=1.0$, (b) $t=1.1$, (c) $t=3.0$, (d) $t=4.1$, (e) $t=7.0$, and (f) $t=10.0$.

Figure 4

Evolution of the the position of the peaks of the histograms shown in Fig. 3, as a function of time. It is clear how these peaks start wide apart and centered around 1 and $-1$. Then, while the behavior of the PDF is that of a bimodal Gaussian distribution, these values fluctuate around $2/\pi$ and $-2/\pi$ (these values are indicated by the horizontal dashed green lines), and finally they vanish as equilibrium is reached in the whole domain.

Figure 5

(a),(b) Evolution of $u(x,t)$ at different times in a simulation with a two-mode initial condition. Two shocks are formed now, and thus two tygers. (c) PDF of $u(x,t=1)$ and proposed PDF for each tyger front. The (green) dashed line in (a) marks the region (before and after the appearance of the shock) contributing to the right peak of the PDF in (c). The mean and standard deviation of this region are the same as the normal distribution plotted with (green) dashed lines in (c). The same applies to other (colored) shaded regions in all three figures. (a) $t=0.4$, (b) $t=1.0$, and (c) $t=1.0$.

Figure 6

Spatio-temporal evolution of $u(x,t)$ from a simulation with a two-mode initial condition: (a) evolution in real space as a function of space and time, and (b) evolution in Fourier space as a function of frequency and wave number. The dashed (blue) lines correspond to the shock velocities, while the solid (green) lines correspond to the tygers. As two shocks form in this case, tygers appear from two different sites. The simple phenomenological theory we present is able to reproduce the mean velocity of each tyger.

Figure 7

Probability density function (PDF) of $u(x,t)$ at $t=3$ for different simulation as (blue) solid lines; each (green) dashed line is the proposed PDF. All simulations have the same initial condition but different resolutions. The bimodal Gaussian distribution plotted is the same for every case. Our results are independent of resolution for the resolutions studied. (a) $N=1024,k_G=341$, (b) $N=2048,k_G=682$, (c) $N=4096,k_G=1365$, (d) $N=8192,k_G=2730$, (e) $N=16384,k_G=5461$, and (f) $N=32768,k_G=10922$.