# Pollution Transport Patterns Obtained Through Generalized Lagrangian Coherent Structures

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Air Quality Model

#### 2.2. Species-Weighted Column-Averaged Generalized Velocity Field

#### 2.3. Eulerian-Lagrangian Analysis

#### 2.4. Computation of Effective Velocity Gradient and Instantaneous Attraction Rate

#### 2.5. Computation of the Lagrangian Trajectory-Averaged Attraction Rate

#### 2.6. Correlation of Column-Averaged Transport Structures with Wind-Only Structures at a Fixed Pressure Layer

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AGL | Above Ground Level |

CMAQ | Community Multiscale Air Quality |

FTLE | Finite-Time Lyapunov Exponent |

LCS | Lagrangian Coherent Structure |

OECS | Objective Eulerian Coherent Structure |

WRF | Weather Research and Forecasting |

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**Figure 1.**Schematic of the effect of a ridgeline of the negative of the attraction rate, $-{s}_{1}^{s}$, on a small parcel of the concentration of species s. ${\mathbf{F}}_{[{t}_{0},t]}^{s}$ is the flow map for species s, as defined in (13).

**Figure 2.**Backward-time FTLE field (relative scale shown in color bar) for the horizontal wind on a pressure layer at approximately 100 m AGL with integration times T of: (

**A**) 0 h, (

**B**) −1 h, and (

**C**) −6 h. See the video version at https://youtu.be/tUZwRAPXAWY.

**Figure 3.**Vertically-integrated species amounts ${Q}^{s}$ (in kg/m${}^{2}$ for H${}_{2}$O and in mg/m${}^{2}$ for other species, as shown in the color bar) superimposed with the species effective flux-based velocity field ${\mathit{\Phi}}^{s}$; the effective velocity vector arrow scale, 10 m/s, is given. The species for each plot are: (

**A**) H${}_{2}$O, (

**B**) SO${}_{2}$, (

**C**) O${}_{3}$, (

**D**) SO${}_{4}^{2-}$, (

**E**) NO${}_{2}$, and (

**F**) Na${}^{+}$.

**Figure 4.**Backward-time FTLE field (relative scale) for the water vapor (H${}_{2}$O) generalized velocity field with integration times T of: (

**A**) 0 h, (

**B**) −1 h, and (

**C**) −6 h. See the video version at https://youtu.be/IKkD9ANZajg. In (

**D**), the correlation coefficient between FTLE field from the species-weighted vertically-integrated velocity field and the FTLE fields for each individual vertical level for integration times of $T=0$ h (blue), $T=-1$ h (orange), and $T=-6$ h (green).

**Figure 10.**The grayscale field is the vertically-integrated species amount ${Q}^{s}$ for $s=$ H${}_{2}$O (top panel), O${}_{3}$ (middle panel), and Na${}^{+}$ (bottom panel). The color bar at the right measures ${Q}^{s}$ in kg/m${}^{2}$ for water vapor H${}_{2}$O and in mg/m${}^{2}$ for ozone O${}_{3}$ and sodium Na${}^{+}$. Superimposed on the ${Q}^{s}$ field are the attractive (blue) and repelling (red) instantaneous hyperbolic Lagrangian coherent structures, material lines within the flow where the attraction and repulsion of pollutant species are maximized. For each species, three snapshots in time are shown, 15:00, 18:00, and 21:00, from left to right.

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**MDPI and ACS Style**

Nolan, P.J.; Foroutan, H.; Ross, S.D. Pollution Transport Patterns Obtained Through Generalized Lagrangian Coherent Structures. *Atmosphere* **2020**, *11*, 168.
https://doi.org/10.3390/atmos11020168

**AMA Style**

Nolan PJ, Foroutan H, Ross SD. Pollution Transport Patterns Obtained Through Generalized Lagrangian Coherent Structures. *Atmosphere*. 2020; 11(2):168.
https://doi.org/10.3390/atmos11020168

**Chicago/Turabian Style**

Nolan, Peter J., Hosein Foroutan, and Shane D. Ross. 2020. "Pollution Transport Patterns Obtained Through Generalized Lagrangian Coherent Structures" *Atmosphere* 11, no. 2: 168.
https://doi.org/10.3390/atmos11020168