Global fluctuations in decaying Burgers turbulence

We consider the statistics of the total energy and the global dissipation in numerical simulations of decaying Burgers turbulence. We observe that at large times these global quantities develop strongly non-Gaussian statistics with asymmetric probability density functions characterized by an exponential tail for values above the mean. Although χ2 distributions would be expected in the description of such definite positive quantities as the energy or the dissipation, we observe that the generalized Gumbel Ga functions provide a better functional form. These distributions have an exponential tail which correctly represents the large fluctuations of the energy or dissipation at large times. We find that the value of the control parameter a(t) of the fitting Gumbel distribution Ga at time t is set by the ratio of the system size to the velocity correlation length, which thus plays the role of an effective number of degrees of freedom. We show that in this problem, the development of exponential tails for the global variables is linked to extreme statistics properties for the largest local events.