Dynamic multiscaling in randomly-forced Burgers turbulence

by Mr Sadhitro De (IISc, Bangalore)

Monday 28 Mar 2022, 15:45 → 17:45 Europe/Stockholm

Albano 3: 6228 - Mega (22 seats) (Albano Building 3)

zoom link : https://stockholmuniversity.zoom.us/j/622224375

 

Most problems of turbulence in geophysical systems like the atmosphere, oceans and the interior of the Earth, to name a few, can be tackled without the need to consider the compressibility of the flow. However, in astrophysical systems such as the interstellar medium and solar atmosphere, shocks play a key role in their dynamics. In this talk, we take a look at the simplest example of shock-dominated turbulence -- the one-dimensional randomly-forced Burgers equation. The nonlinearity of the Burgers equation is akin to that of the Navier-Stokes equation due to which it is often sought as a testing ground for statistical theories of turbulence. Furthermore, the Burgers equation can be derived by taking a spatial derivative Kardar-Parisi-Zhang equation which is used as a model for growing interfaces. With smooth initial conditions and/or forcing, the Burgers equation can be exactly solved via the Hopf-Cole transformation. It is well-known from both theoretical arguments and numerical simulations that for a Gaussian, white-in-time random forcing whose power spectrum scales with the wave-number, k, as 1/k, the one-dimensional Burgers velocity profile reaches a statistically stationary state with (a) an energy spectrum which scales as  k^{-5/3} (same as that predicted by Kolmogorov's theory of turbulence) and, (b) bifractal fluctuations due to the presence of shocks. As time progresses, Lagrangian intervals collapse to points at shocks which are regions of high flow convergence. Hence, a natural time-scale to define here is the time, \tau, taken by an interval of a certain length, r, to cvollapse at a shock. We call \tau the time of collapse. Through high-resolution direct numerical simulations of our model and detailed theoretical analysis, we show that the statistics of \tau are highly non-Gaussian. The p-th order moment of \tau scales as a power-law with respect to r with the power-law exponent, z_p. We refer to this dynamic exponent as the collapse-time exponent. Different values of z_p are required to characterize the scaling of the moments of different orders p. Such existence of multiple dynamic scaling exponents implies dynamic multiscaling. We propose generalization of our results to higher dimensions.