# Data from: Understanding density fluctuations in supersonic, isothermal turbulence

## Abstract

Supersonic turbulence occurs in many environments, particularly in astrophysics. In the crucial case of isothermal turbulence, the probability density function (PDF) of the logarithmic density, s, is well-measured, but a theoretical understanding of the processes leading to this distribution remains elusive. We investigate these processes using Lagrangian tracer particles to track s and ds/dt in direct numerical simulations, and we show that their evolution can be modeled as a stochastic differential process with time-correlated noise. The temporal correlation functions of s and ds/dt decay exponentially, as predicted by the model, and the decay timescale is $\approx$ 1/6 the eddy turnover time. The behavior of the conditional averages of ds/dt and d2s/dt2 is also well explained by the model, which shows that the density PDF arises from a balance between stochastic compressions/expansions, which tend to broaden the PDF, and the acceleration/deceleration of shocks by density gradients, which tends to narrow it.

## README: Data for Understanding Density Fluctuations in Supersonic, Isothermal Turbulence

https://doi.org/10.5061/dryad.zgmsbccn0

### Description of the data and file structure

A repository of the reduced data and macros used to generate the main results of this paper.

#### Files and variables

##### File: Data.zip

**Description:** A zipped repository of the reduced simulation data used to generate the results in this paper. This includes temporal correlation function data from each of the runs, along with value of ds/dt and d^2s/dt^2 as a function of s. The data also computes particle values interpolated by CIC, which is shown to be the best method in the paper, as well as a quadratic method, which is shown to be less reliable. Finally, the repository includes macros for reproducing the main results in the paper.

### Files and Directory Structure

Each directory is named according to its name in the corresponding paper. The naming scheme is determined by Mach number and viscosity. Each directory contains the same files, except for the _256 and _718, which contain a subset of the files relevant to the convergence study presented in the appendix.

#### Corr.csv

This file contains the correlation function

- Column 1 – log 10 time in units of L_box/cs
- Column 2 – Correlation function of s as measured by the CIC method
- Column 3 – Correlation function of s as measured by the Quadratic method
- Column 4 – Correlation function of ds/dt as measured by the CIC method
- Column 5 – Correlation function of ds/dt as measured by the Quadratic method (time in L_box/cs)

#### Dsdt_cond_C_mean.csv

Gives values of various quantities conditional on s as measured by the particles using the CIC method

- Column 1 – This is the center of the s bin
- Column 2 – This is the width of the s bin
- Column 3 – This is the number of particles with negative ds/dt in this s bin
- Column 4 – This is the number of particles with positive ds/dt in this s bin
- Column 5 – This is the sum of ds/dt for all particles with negative ds/dt in this s bin
- Column 6 – This is the sum of ds/dt for all particles with positive ds/dt in this s bin
- Column 7 – This is the sum of (ds/dt)^2 for all particles with negative ds/dt in this s bin
- Column 8 – This is the sum of (ds/dt)^2 for all particles with positive ds/dt in this s bin
- Column 9 – This is the sum of d^2s/dt^2 for all particles with negative ds/dt in this s bin
- Column 0 – This is the sum of d^2s/dt^2 for all particles with positive ds/dt in this s bin

#### Dsdt_cond_Q_mean.csv

Gives values of various quantities conditional on s as measured by the particles using the Quadratic method, columns are as in Dsdt_cond_C_mean.csv

#### Dsdt_cond_V_mean.csv

Gives values of various quantities conditional on s as measured using the divergence as computed from the grid instead of ds/dt from the particles

- Column 1 – This is the center of the s bin
- Column 2 – This is the width of the s bin
- Column 3 – This is the number of particles with negative ds/dt in this s bin
- Column 4 – This is the number of particles with positive ds/dt in this s bin
- Column 5 – This is the sum of ds/dt for all particles with negative ds/dt in this s bin
- Column 6 – This is the sum of ds/dt for all particles with positive ds/dt in this s bin

#### S_C_mean.csv

Gives histogram of s as measured from the particles using the CIC method

- Column 1 – This is the center of the s bin
- Column 2 – This is the width of the s bin
- Column 3 – This value for the bin

#### S_Q_mean.csv

Gives histogram of s as measured from the particles using the Quatratic method, Columns are as in S_C_mean.csv

#### S_rho_mean.csv

Gives the mass-weighted of s as measured from the grid. Columns are as in S_C_mean.csv

#### Dsdt_C_mean.csv

Gives histogram of ds/dt as measured from the particles using the CIC method

- Column 1 – This is the center of the ds/dt bin for the negative part of the histogram
- Column 2 – This is the width of the ds/dt bin or the negative part of the histogram
- Column 3 – This is the value for the bin
- Column 4 – This is the center of the ds/dt bin for the positive part of the histogram
- Column 5 – This is the width of the ds/dt bin or the positive part of the histogram
- Column 6 – This is the value for the bin

#### Dsdt_Q_mean.csv

Gives histogram of ds/dt as measured from the particles using the quatratic method, columns are as in Dsdt_C_mean.csv

#### Dsdt_V_mean.csv

Gives histogram of ds/dt as measured from the divergence from the grid, columns are as in Dsdt_C_mean.csv

### Access information

Other publicly accessible locations of the data:

- None

Data was derived from the following sources:

- FLASH code

## Methods

Generated with the FLASH code and reduced as described in the paper.