# Baropycnal Work: A Mechanism for Energy Transfer across Scales

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (I)
- Barotropic and baroclinic generation of strain, $\mathit{S}$, from gradients of pressure and density, $\rho $:$\left(\mathrm{const}.\right)\phantom{\rule{0.166667em}{0ex}}{\ell}^{2}\phantom{\rule{0.166667em}{0ex}}{\rho}^{-1}\left(\right)open="["\; close="]">\mathbf{\nabla}P\mathbf{\xb7}\mathbf{S}\mathbf{\xb7}\mathbf{\nabla}\rho $,
- (II)
- Baroclinic generation of vorticity, $\mathit{\omega}$:$\left(\mathrm{const}.\right)\phantom{\rule{0.166667em}{0ex}}{\ell}^{2}\phantom{\rule{0.166667em}{0ex}}{\rho}^{-1}\left(\right)open="("\; close=")">\mathbf{\nabla}\rho \times \mathbf{\nabla}P$,

## 2. Multi-Scale Dynamics

#### Variable Density Flows

## 3. The Mechanism of Baropycnal Work

## 4. Simulations

## 5. Numerical Results

## 6. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Momentum magnitude $\left|\rho \mathbf{u}\right|$ from (

**a**) Run 1 ($\zeta =0.01$) and (

**b**) Run 3 ($\zeta =0.6$).

**Figure 3.**Spectra of velocity (u) and its dilatational and solenoidal (${u}^{d}$ and ${u}^{s}$, respectively) components from Run 1. The reference dashed black lines have slopes of $-5/3$ and $-2$.

**Figure 4.**Flux terms ${\Pi}_{\ell}$ and ${\Lambda}_{\ell}$ from Run 1, as well as their sum averaged over space and time, as a function of the filtering wavenumber $k=2\pi /\ell $. Filtering here uses the sharp-spectral filter kernel. The star superscript indicates normalization by the effective kinetic energy injection, ${\epsilon}^{eff}={\epsilon}^{inj}+\langle p\mathbf{\nabla}\mathbf{\xb7}\mathbf{u}\rangle $.

**Figure 5.**Correlation between baropycnal work and its nonlinear model from Run 1 using the box kernel with filter scale ${k}_{\ell}=8$ in (

**a**,

**c**) and ${k}_{\ell}=16$ in (

**b**,

**d**). Top two panels plot $\Lambda $ and ${\Lambda}_{m}$ along a diagonal line through the domain from a single snapshot. Lower two panels show time-averaged isocontours of the logarithm of the joint PDF between $\Lambda $ and ${\Lambda}_{m}$, where star superscripts indicate that means have been subtracted and the values are normalized by their variance. Straight-red lines are $y=x$. The correlation coefficients are ${R}_{c}=0.93$ at filter scale ${k}_{\ell}=8$ and ${R}_{c}=0.94$ at ${k}_{\ell}=16$. All four panels indicate excellent correlation between $\Lambda $ and ${\Lambda}_{m}$.

**Figure 6.**Same as in Figure 5 but using the Gaussian filter. The correlation coefficients are ${R}_{c}=0.97$ at filter scale ${k}_{\ell}=8$ and ${R}_{c}=0.97$ at ${k}_{\ell}=16$. All four panels indicate excellent correlation between $\Lambda $ and ${\Lambda}_{m}$.

**Figure 7.**Same as in Figure 5 but using the sharp-spectral filter. The correlation coefficients are ${R}_{c}=0.27$ at filter scale ${k}_{\ell}=8$ and ${R}_{c}=0.28$ at ${k}_{\ell}=16$. The correlation between $\Lambda $ and ${\Lambda}_{m}$ is poor when using a sharp-spectral filter due to its nonpositivity, which can yield negative filtered densities [15] and physically unrealizable stresses [46].

**Figure 8.**Pointwise comparison between (

**a**) baropycnal work and (

**b**) its nonlinear model from a 2D slice of the 3D domain in Run 1, at one instant in time. A Gaussian kernel at scale ${k}_{\ell}=8$ is used. The visualizations show excellent pointwise correlation.

**Table 1.**Comparison of compressibility metrics and cascade terms at different grid resolution. Low-$\zeta $ corresponds to high compressibility in the external forcing. The spatially averaged cascade terms $\langle {\Lambda}_{\ell}\rangle $ and $\langle {\Pi}_{\ell}\rangle $ are calculated using the sharp-spectral cutoff filter with ${k}_{\ell}=2\pi /\ell =6$. $\frac{\Delta x}{\eta}$ is the ratio of grid size to the Kolmogorov length.

Run | N | $\mathit{\zeta}$ | ${\mathit{M}}_{\mathit{t}}$ | ${\mathit{Re}}_{\mathit{\lambda}}$ | $\frac{{(\mathit{\nabla}\mathbf{\xb7}\mathbf{u})}_{\mathbf{rms}}}{{(\mathit{\nabla}\times \mathbf{u})}_{\mathbf{rms}}}$ | $\frac{{\mathbf{k}}^{\mathit{d}}}{{\mathbf{k}}^{\mathit{s}}}$ | $\frac{\mathbf{\Delta}\mathbf{x}}{\mathit{\eta}}$ | $\left(\right)open="\langle "\; close="\rangle ">{\mathbf{\Pi}}_{\mathit{\ell}}$ | $\left(\right)open="\langle "\; close="\rangle ">{\mathbf{\Lambda}}_{\mathit{\ell}}$ |
---|---|---|---|---|---|---|---|---|---|

1 | 1024 | 0.01 | 0.23 | 65 | 0.50 | 0.74 | 0.23 | $8.9\times {10}^{-3}$ | $-5.6\times {10}^{-3}$ |

2 | 512 | 0.01 | 0.22 | 33 | 0.46 | 0.51 | 0.28 | $6.8\times {10}^{-3}$ | $-4.8\times {10}^{-3}$ |

3 | 512 | 0.6 | 0.33 | 206 | 0.05 | 0.02 | 1.78 | $2.3\times {10}^{-2}$ | $-9.6\times {10}^{-5}$ |

4 | 256 | 0.01 | 0.21 | 18 | 0.54 | 0.56 | 0.25 | $3.6\times {10}^{-3}$ | $-3.1\times {10}^{-3}$ |

5 | 256 | 0.6 | 0.42 | 150 | 0.04 | 0.01 | 2.1 | $5.0\times {10}^{-2}$ | $-6.0\times {10}^{-4}$ |

6 | 256 | 1.0 | 0.46 | 175 | 0.03 | 0.003 | 2.2 | $5.0\times {10}^{-2}$ | $-3.8\times {10}^{-4}$ |

7 | 128 | 0.01 | 0.20 | 10 | 0.65 | 0.80 | 0.24 | $8.0\times {10}^{-4}$ | $-7.5\times {10}^{-4}$ |

8 | 128 | 0.6 | 0.50 | 105 | 0.05 | 0.01 | 2.3 | $4.0\times {10}^{-2}$ | $-4.0\times {10}^{-4}$ |

9 | 128 | 1.0 | 0.40 | 95 | 0.03 | 0.01 | 2.0 | $2.5\times {10}^{-2}$ | $-2.0\times {10}^{-4}$ |

**Table 2.**The types of filters used in calculating $\Lambda $ and ${\Lambda}_{m}$. The Heaviside function $H\left(x\right)=1$ for $x\ge 0$ and $H\left(x\right)=0$ for $x<0$. The correlation coefficient ${R}_{c}$ is shown at two scales ${k}_{\ell}=2\pi /\ell $.

Filter Type | Kernel | ${\mathit{R}}_{\mathit{c}}|\phantom{\rule{0.166667em}{0ex}}{\mathbf{k}}_{\mathit{\ell}}=8$ | ${\mathit{R}}_{\mathit{c}}|\phantom{\rule{0.166667em}{0ex}}{\mathbf{k}}_{\mathit{\ell}}=16$ |
---|---|---|---|

Box | ${G}_{\ell}\left(\mathbf{x}\right)={\prod}_{i=1}^{3}\frac{1}{\ell}H\left(\right)open="("\; close=")">\frac{\ell}{2}-\left(\right)open="|"\; close="|">{x}_{i}$ | 0.93 | 0.94 |

Gaussian | ${G}_{\ell}\left(\mathbf{x}\right)=\frac{1}{{\ell}^{3}}{\left(\right)}^{\frac{6}{\pi}}3/2$ | 0.97 | 0.97 |

Sharp spectral | ${\widehat{G}}_{\ell}\left(\mathbf{k}\right)={\prod}_{i=1}^{3}H\left(\right)open="("\; close=")">\frac{2\pi}{\ell}-\left(\right)open="|"\; close="|">{k}_{i}$ | 0.27 | 0.28 |

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Lees, A.; Aluie, H.
Baropycnal Work: A Mechanism for Energy Transfer across Scales. *Fluids* **2019**, *4*, 92.
https://doi.org/10.3390/fluids4020092

**AMA Style**

Lees A, Aluie H.
Baropycnal Work: A Mechanism for Energy Transfer across Scales. *Fluids*. 2019; 4(2):92.
https://doi.org/10.3390/fluids4020092

**Chicago/Turabian Style**

Lees, Aarne, and Hussein Aluie.
2019. "Baropycnal Work: A Mechanism for Energy Transfer across Scales" *Fluids* 4, no. 2: 92.
https://doi.org/10.3390/fluids4020092