# A Budget-Based Turbulence Length Scale Diagnostic

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework

#### 2.1. Spectral TKE Equation

#### 2.2. Formulation of the New Turbulence Length Scale

## 3. Method and Data

#### 3.1. Estimation of the New Turbulence Length Scale

#### 3.2. LES Simulations

#### 3.3. Fit of Turbulence Length Scale

#### 3.4. Computation of Turbulence Fluxes

#### 3.5. Gravity Waves

## 4. Results

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ABL | Atmospheric Boundary Layer |

ARM | Atmospheric Radiation Measurement |

BOMEX | Barbados Oceanographic and Meteorological Experiment |

DYCOMS-II | second Dynamics and Chemistry of Marine Stratocumulus |

GABLS | GEWEX Atmospheric Boundary Layer Study |

GC | Global Circulation |

LES | Large Eddy Simulation |

NWP | Numerical Weather Prediction |

TKE | Turbulence Kinetic Energy |

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**Figure 1.**The TKE budget terms (

**a**,

**d**,

**g**,

**j**), the TKE (

**b**,

**e**,

**h**,

**k**) and the normalized differences in the effective dissipation rate, $\Delta \tilde{{\u03f5}_{c}}\left(\right)open="("\; close=")">{l}_{c}/{max}_{z}\left(\u03f5\right)$, (

**c**,

**f**,

**i**,

**l**) for four cases: ARM (first row), BOMEX (second row), DYCOMS-II (third row), and GABLS1 (fourth row). The TKE budget terms are displayed for the smallest sub-domain size, where side of the sub-domain is equal to 1/32 of the side of the whole domain. The TKE and the differences in the effective dissipation rate are displayed for different sub-domain side sizes, ${l}_{c}$.

**Figure 3.**As in Figure 1, but comparison of the longitudinal integral turbulence length scales (see Equation (25)) for the zonal wind, ${L}_{i}^{u,x}$, and the meridional wind, ${L}_{i}^{v,y}$ is computed for the period one hour before the end of the simulation (

**a**,

**d**,

**g**,

**j**); diagnosed turbulence length scales computed from the effective dissipation rate, ${L}_{c}\left(\right)open="("\; close=")">{l}_{c}$ (see Equation (23)); and the dissipation rate, ${L}_{\u03f5}\left(\right)open="("\; close=")">{l}_{c}$ (see Equation (24)), (

**b**,

**e**,

**h**,

**k**); and their respective fits with the relationship in Equation (26) (

**c**,

**f**,

**i**,

**l**). The parametrized turbulence length scale according to Equation (18) in [16], ${L}_{B}$, is also presented (

**c**,

**f**,

**i**,

**l**). The fits for the ARM and BOMEX cases are done only with data below the cloud top (indicated by horizontal line).

**Figure 4.**The turbulent heat flux (

**a**) and the turbulent momentum fluxes (

**b**,

**c**) for GABLS1. The turbulent fluxes are computed according to the local down-gradient formulation (see Equations (27)–(29)) using the fit of the diagnosed turbulent length scale, ${L}_{c}$, (cyan lines), fit of ${L}_{\u03f5}$ (green lines), or the parametrized turbulence length scale according to Equation (18) in [16], ${L}_{B}$, (red lines) for two sub-domain sizes (dashed lines versus solid lines). The turbulent fluxes acquired directly from LES are plotted with blue lines.

**Table 1.**Horizontal and vertical domain size, horizontal and vertical resolution, and integration time for the LES runs.

Case | Hor. Domain Size | Hor. Resol. | Ver. Domain Size | Ver. Resol. | Integration Time |
---|---|---|---|---|---|

ARM | 12.8 km × 12.8 km | 12.5 m | 1500 m | 31.125 m | 10 h |

BOMEX | 12.8 km × 12.8 km | 12.5 m | 3000 m | 23.44 m | 6 h |

DYCOMS-II | 12.8 km × 12.8 km | 12.5 m | 1500 m | 2.93 m | 4 h |

GABLS1 | 800 m × 800 m | 0.78125 m | 400 m | 0.78125 m | 6 h |

**Table 2.**First guess and bounds for the fitting constants in Equation (26).

Fitting Constant | First Guess | Minimal Value | Maximal Value |
---|---|---|---|

${H}_{\mathrm{abl}}$ | 1000 m | 0 m | 2000 m |

${\lambda}_{m}$ | 350 m | 0 m | 1000 m |

${a}_{m}$ | 5.5 | 0 | 100 |

${b}_{m}$ | 3.0 | 0 | 100 |

${\beta}_{m}$ | 0.02 | −0.2 | 0.5 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Bašták Ďurán, I.; Schmidli, J.; Bhattacharya, R.
A Budget-Based Turbulence Length Scale Diagnostic. *Atmosphere* **2020**, *11*, 425.
https://doi.org/10.3390/atmos11040425

**AMA Style**

Bašták Ďurán I, Schmidli J, Bhattacharya R.
A Budget-Based Turbulence Length Scale Diagnostic. *Atmosphere*. 2020; 11(4):425.
https://doi.org/10.3390/atmos11040425

**Chicago/Turabian Style**

Bašták Ďurán, Ivan, Juerg Schmidli, and Ritthik Bhattacharya.
2020. "A Budget-Based Turbulence Length Scale Diagnostic" *Atmosphere* 11, no. 4: 425.
https://doi.org/10.3390/atmos11040425