Anomaly and Brownian fluid particle in Navier-Stokes turbulence

We investigate the Navier-Stokes turbulence driven by a stochastic random Gaussian force. Using a field-theoretic approach, we uncover an anomaly that brings hidden structure to the theory. The anomaly is generated by a non-self-adjoint operator of the Jacobian and it follows the symmetries of the stochastic Navier-Stokes equation. We calculate the anomaly and demonstrate that by forcing the anomaly to vanish, the velocity field is constrained and a monopole-type object with a constant charge is formed. When the viscosity is zero, the anomaly can be interpreted as the Brownian damping coefficient of a random fluid particle. We provide the Brownian particle equation and its solution in the presence of a pump and viscosity. Our results suggest that the anomaly is an inherent feature of stochastic turbulence and must be taken into account in all stochastic turbulence calculations. This constitutes an additional law for the original set of stochastic Navier-Stokes equations.Comment: 16 pages. One more additio

Local symmetries, anomalies and constrains in Burgers Turbulence

We study stochastic Burgers turbulence without pressure. We first show that the variational derivative of the Burgers equation is dependent on the velocity field, suggesting the existence of an anomaly. The anomaly is created by an operator that is non-self-adjoint. To calculate it correctly, we need to find its square. There are similarities with conformal and chiral two-dimensional field theories, but causality is the key that makes the difference. We find a local symmetry for the Burgers equation that is broken by the anomaly. By requiring the disappearance of this anomaly, the velocity field is constrained and local symmetry is maintained. This symmetry follows Kolmogorov's second law of self-similarity. One can choose an anomaly-free theory, a partially broken theory, or a fully broken theory by choosing the constraint appropriately. There is an analogy to gauge fixing or vacuum selection which define the local configuration.Comment: 10 pages. Details added and typos correcte