Spatial-temporal structure functions in Burgers turbulence driven by an Ornstein-Uhlenbeck process

TL;DR

This paper derives and verifies an expression for the spatial-temporal structure functions in Burgers turbulence driven by an Ornstein-Uhlenbeck process, revealing exponential decay and differences from white-noise-driven turbulence.

Contribution

It provides a new analytical expression for the third-order spatial-temporal structure function in OU-driven Burgers turbulence and compares its properties with white-noise-driven turbulence.

Findings

High-order structure functions decay exponentially with different rates for odd and even orders.

Spatial structure functions are identical for OU-driven and white-noise-driven turbulence at the same energy injection rate.

Velocity PDF in OU-driven turbulence is bimodal, unlike the near-Gaussian distribution in white-noise-driven turbulence.

Abstract

We explore the spatial-temporal structure functions of Burgers turbulence driven by a temporal Ornstein-Uhlenbeck (OU) process, where the characteristic time scale of the OU process is much larger than that of the energy flux across spatial scales. Based on the K\'arm\'an-Howarth-Monin equation, we obtain an expression for the third-order spatial-temporal structure function away from the dissipation scale. This expression combines Kolmogorov's exact result of spatial structure function and the exponential temporal decay of the external force. We numerically justify this expression and find that the high-order structure functions also decay exponentially, however, the dependence of decay rates on order is different for the odd- and even-order structure functions. Comparing the OU-driven Burgers turbulence with that driven by temporal white noise, their spatial structure functions are identical when the energy injection rates are the same, which justifies Kolmogorov's theory, but these two systems' temporal structure functions differ. Also, the velocity pdf in the OU-driven Burgers turbulence shows a bimodal distribution, contradicting the near-Gaussian distribution in white-noise-driven turbulence.