# Asymptotic Theory for the Probability Density Functions in Burgers Turbulence

## Abstract

A systematic analysis is carried out for the randomly forced Burgers equation in the infinite Reynolds number (inviscid) limit. No closure approximations are made. Instead the probability density functions of velocity and velocity gradient are related to the statistics of quantities defined along the shocks. This method allows one to compute the dissipative anomalies, as well as asymptotics for the structure functions and the probability density functions. It is shown that the left tail for the probability density function of the velocity gradient has to decay faster than {vert_bar}{xi}{vert_bar}{sup {minus}3} . A further argument confirms the prediction of E {ital et al.}thinspthinsp[Phys.thinspthinspRev.thinspthinspLett.thinspthinsp{bold 78}, 1904 (1997)] that it should decay as {vert_bar}{xi}{vert_bar}{sup {minus}7/2} . {copyright} {ital 1999} {ital The American Physical Society}

- Authors:

- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 (United States)

- Publication Date:

- OSTI Identifier:
- 692514

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review Letters

- Additional Journal Information:
- Journal Volume: 83; Journal Issue: 13; Other Information: PBD: Sep 1999

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 66 PHYSICS; PROBABILITY; VELOCITY; TURBULENCE; DISTRIBUTION FUNCTIONS; STOCHASTIC PROCESSES; ASYMPTOTIC SOLUTIONS; SHOCK WAVES

### Citation Formats

```
E, W., and Vanden Eijnden, E.
```*Asymptotic Theory for the Probability Density Functions in Burgers Turbulence*. United States: N. p., 1999.
Web. doi:10.1103/PhysRevLett.83.2572.

```
E, W., & Vanden Eijnden, E.
```*Asymptotic Theory for the Probability Density Functions in Burgers Turbulence*. United States. doi:10.1103/PhysRevLett.83.2572.

```
E, W., and Vanden Eijnden, E. Wed .
"Asymptotic Theory for the Probability Density Functions in Burgers Turbulence". United States. doi:10.1103/PhysRevLett.83.2572.
```

```
@article{osti_692514,
```

title = {Asymptotic Theory for the Probability Density Functions in Burgers Turbulence},

author = {E, W. and Vanden Eijnden, E.},

abstractNote = {A systematic analysis is carried out for the randomly forced Burgers equation in the infinite Reynolds number (inviscid) limit. No closure approximations are made. Instead the probability density functions of velocity and velocity gradient are related to the statistics of quantities defined along the shocks. This method allows one to compute the dissipative anomalies, as well as asymptotics for the structure functions and the probability density functions. It is shown that the left tail for the probability density function of the velocity gradient has to decay faster than {vert_bar}{xi}{vert_bar}{sup {minus}3} . A further argument confirms the prediction of E {ital et al.}thinspthinsp[Phys.thinspthinspRev.thinspthinspLett.thinspthinsp{bold 78}, 1904 (1997)] that it should decay as {vert_bar}{xi}{vert_bar}{sup {minus}7/2} . {copyright} {ital 1999} {ital The American Physical Society}},

doi = {10.1103/PhysRevLett.83.2572},

journal = {Physical Review Letters},

number = 13,

volume = 83,

place = {United States},

year = {1999},

month = {9}

}