# Time-Reversibility in Atmospheric Dispersion

## Abstract

**:**

“When you stir your rice pudding, Septimus, the spoonful of jam spreads itself round making red trails like the picture of a meteor in my astronomical atlas. But if you stir backwards, the jam will not come together again. Indeed, the pudding does not notice and continues to turn pink just as before. Do you think this is odd?”Tom Stoppard: Arcadia

## 1. Introduction

**Figure 1.**Forward (black) and backward (colored) pollutant clouds emitted instantaneously and consisting of ${n}_{0}=1.25\times {10}^{5}$ particles at (

**a**) $t=0$ and (

**b**) $t=7$ days. The forward pollutant cloud is initiated on 12 March 2011 at 00 UTC at ${120}^{\circ}$E, ${20}^{\circ}$N, 500 hPa in a ${1}^{\circ}\times {1}^{\circ}\times 100$ hPa cuboid beyond Indonesia. Colored dots denote the location of the particles whose backward simulation started at time ${t}_{\mathrm{b}}=2$ days (blue), ${t}_{\mathrm{b}}=7$ days (yellow) and ${t}_{\mathrm{b}}=9$ days (red), respectively. In the inset of panel (a) instead of black dots the initial position of the particles is illustrated by a black rectangular frame. This simulation is an element of the global study in Section 6.

**Figure 2.**An outline of the time schedule of the forward and backward simulations. Forward simulation starts at Time 0 and terminates after nine days. Backward simulations start at time ${t}_{\mathrm{b}}$ (where the value of ${t}_{\mathrm{b}}$ can be $1,2,\dots ,9$ days) and terminate at Time 0. The comparison of the forward and backward cloud is carried out at time t ($t=0,1,\dots ,{t}_{\mathrm{b}}-1$ days) as a function of the time period $\Delta {t}_{\mathrm{b}}$ of the backward tracking.

## 2. The RePLaT Dispersion Model

## 3. Input Data

## 4. Statistical Parameters

#### 4.1. Proportion of Particles Returned to the Initial Volume

#### 4.2. Center of Mass and Standard Deviation of the Pollutant Clouds

#### 4.3. Figure of Merit in Space

#### 4.4. Pearson’s Correlation Coefficient

## 5. Case Study

**Figure 3.**Forward (black) and backward (colored) pollutant clouds emitted instantaneously and consisting of ${n}_{0}=1.25\times {10}^{5}$ particles of the case study at (

**a**) $t=0$, (

**b**) $t=2$, (

**c**) $t=5$ and (

**d**) $t=7$ days. The forward pollutant cloud is initiated on 12 March 2011 at 00 UTC at ${141}^{\circ}$E, $37.{5}^{\circ}$N, 500 hPa in a ${1}^{\circ}\times {1}^{\circ}\times 100$ hPa cuboid beyond Fukushima. Dots colored according to the color bar on the right denote the location of the particles whose backward simulation started at time ${t}_{\mathrm{b}}$ (day). In the inset of panel (a), instead of black dots, the initial position of the particles is illustrated by a black rectangular frame. For better visibility, only every fifth particle of the clouds is plotted.

**Figure 4.**(

**a**) The horizontal standard deviation ${\sigma}_{\mathrm{h},\mathrm{f}}$ of the forward cloud (FWC) in Figure 3 as a function of time t with an exponential fit over $t=0$–2 days (dashed line) and a power-law fit over $t=2$–9 days (dash-dotted line); (

**b**) the horizontal standard deviation ${\sigma}_{\mathrm{h},\mathrm{b}}$ of the backward clouds (BWCs) in Figure 3 as a function of the time interval of the backward simulation $\Delta {t}_{\mathrm{b}}={t}_{\mathrm{b}}-t$. Solid lines represent the time evolution of the BWCs initiated at ${t}_{\mathrm{b}}$ (see color legend) backward in time; dashed lines connect the values corresponding to time t.

**Figure 5.**(

**a**) The horizontal difference $\Delta {\mathrm{CM}}_{\mathrm{h}}$ between the center of mass of the BWCs and the FWC depending on the time period $\Delta {t}_{\mathrm{b}}$ of the backward simulation; (

**b**) the difference $\Delta {\sigma}_{\mathrm{h}}$ between the horizontal standard deviations of the BWCs and FWC; (

**c**) the proportion $n/{n}_{0}$ of the particles returned back to the initial cuboid. Colored triangles indicate the starting time ${t}_{\mathrm{b}}$ of the backward simulations (see color legend); black diamonds (in some cases in overlap with triangles) are the mean values belonging to a given $\Delta {t}_{\mathrm{b}}$ (denoted by “m”). Dashed lines indicate exponential fits to the mean values; the exponents are given in Table 1. The dash-dotted line in panel (b) indicates a power-law fit to mean values over the interval $\Delta {t}_{\mathrm{b}}=3$–9 days with an exponent of 4.3.

**Table 1.**The exponent Λ (day${}^{-1}$) and the error of the exponential fits to the curves $\Delta {\mathrm{CM}}_{\mathrm{h}}$, $\Delta {\sigma}_{\mathrm{h}}$, $n/{n}_{0}$, ${\mathrm{FMS}}_{2\mathrm{D}}$, ${\mathrm{FMS}}_{3\mathrm{D}}$, ${\mathrm{PCC}}_{2\mathrm{D}}$ and ${\mathrm{PCC}}_{3\mathrm{D}}$ versus $\Delta {t}_{\mathrm{b}}$ of the case study in Figure 5 and Figure 6.

- | Λ |
---|---|

$\Delta {\mathrm{CM}}_{\mathrm{h}}$ | $0.481\pm 0.032$ |

$\Delta \sigma $ | $0.467\pm 0.050$ |

$n/{n}_{0}$ | $0.339\pm 0.032$ |

${\mathrm{FMS}}_{2\mathrm{D}}$ | $0.466\pm 0.048$ |

${\mathrm{FMS}}_{3\mathrm{D}}$ | $0.456\pm 0.040$ |

${\mathrm{PCC}}_{2\mathrm{D}}$ | $0.447\pm 0.028$ |

${\mathrm{PCC}}_{3\mathrm{D}}$ | $0.437\pm 0.027$ |

**Figure 6.**(

**a**) The figure of merit in space ${\mathrm{FMS}}_{3\mathrm{D}}$ between the BWCs and FWC depending on the time period $\Delta {t}_{\mathrm{b}}$ of the backward simulation; (

**b**) Pearson’s correlation coefficient ${\mathrm{PCC}}_{3\mathrm{D}}$ between the BWCs and FWC. Colored triangles indicate the starting time ${t}_{\mathrm{b}}$ of the backward simulations (see the color legend); black diamonds (in some cases in overlap with triangles) are the mean values belonging to a given $\Delta {t}_{\mathrm{b}}$ (denoted by “m”). Dashed lines indicate exponential fits to mean values; the exponents are given in Table 1.

## 6. Global Results

**Figure 7.**(

**a**) The horizontal difference $\Delta {\mathrm{CM}}_{\mathrm{h}}$ between the center of mass of the BWCs and the FWC depending on the time period $\Delta {t}_{\mathrm{b}}$ of the backward simulation; (

**b**) the difference $\Delta {\sigma}_{\mathrm{h}}$ between the horizontal standard deviations of the BWCs and FWC; (

**c**) the proportion $n/{n}_{0}$ of the particles returned back to the initial cuboid. The results are shown for the tropical region (TR, red), for mid- and high latitudes (M/H, blue) and for the globe (GL, black). Mean values (diamonds), medians and lower and upper quartiles (boxes with horizontal lines) are indicated. Dashed lines indicate exponential fits to the mean values; the exponents are given in Table 2.

**Figure 8.**(

**a**) The figure of merit in space ${\mathrm{FMS}}_{3\mathrm{D}}$ between the BWCs and FWC depending on the time period $\Delta {t}_{\mathrm{b}}$ of the backward simulation; (

**b**) Pearson’s correlation coefficient ${\mathrm{PCC}}_{3\mathrm{D}}$ between the BWCs and FWC. Mean values (diamonds), medians and lower and upper quartiles (boxes with horizontal lines) are indicated for the tropical region (TR, red), for mid- and high latitudes (M/H, blue) and for the globe (GL, black). Dashed lines indicate exponential fits to the mean values; the exponents are given in Table 2.

**Table 2.**The exponent Λ (day${}^{-1}$) and the error of the exponential fits to the curves $\Delta {\mathrm{CM}}_{\mathrm{h}}$, $\Delta {\sigma}_{\mathrm{h}}$, $n/{n}_{0}$, ${\mathrm{FMS}}_{2\mathrm{D}}$, ${\mathrm{FMS}}_{3\mathrm{D}}$, ${\mathrm{PCC}}_{2\mathrm{D}}$ and ${\mathrm{PCC}}_{3\mathrm{D}}$ versus $\Delta {t}_{\mathrm{b}}$ for the globe (GL), for the tropical region (TR) and for mid- and high latitudes (M/H) in Figure 7 and Figure 8.

- | GL | TR | M/H |
---|---|---|---|

$\Delta {\mathrm{CM}}_{\mathrm{h}}$ | $0.290\pm 0.015$ | $0.267\pm 0.020$ | $0.422\pm 0.013$ |

$\Delta \sigma $ | $0.358\pm 0.035$ | $0.330\pm 0.037$ | $0.495\pm 0.034$ |

$n/{n}_{0}$ | $0.202\pm 0.001$ | $0.122\pm 0.006$ | $0.293\pm 0.004$ |

${\mathrm{FMS}}_{2\mathrm{D}}$ | $0.314\pm 0.025$ | $0.224\pm 0.031$ | $0.407\pm 0.031$ |

${\mathrm{FMS}}_{3\mathrm{D}}$ | $0.311\pm 0.014$ | $0.217\pm 0.020$ | $0.417\pm 0.019$ |

${\mathrm{PCC}}_{2\mathrm{D}}$ | $0.300\pm 0.058$ | $0.212\pm 0.064$ | $0.404\pm 0.070$ |

${\mathrm{PCC}}_{3\mathrm{D}}$ | $0.281\pm 0.029$ | $0.196\pm 0.036$ | $0.390\pm 0.040$ |

## 7. Outlook

**Table 3.**The exponent Λ (day${}^{-1}$) and the error of the exponential fits to the curves $\Delta {\mathrm{CM}}_{\mathrm{h}}$, $\Delta {\sigma}_{\mathrm{h}}$, $n/{n}_{0}$, ${\mathrm{FMS}}_{2\mathrm{D}}$, ${\mathrm{FMS}}_{3\mathrm{D}}$, ${\mathrm{PCC}}_{2\mathrm{D}}$ and ${\mathrm{PCC}}_{3\mathrm{D}}$ versus $\Delta {t}_{\mathrm{b}}$ of the case study described in Section 5, but with horizontal turbulent diffusivity ${K}_{\mathrm{h}}=50$ m${}^{2}$·s${}^{-1}$. The deviation $\Delta \Lambda $ of the Λ values of the case study with turbulent diffusivity from the case study without turbulent diffusivity is also shown.

- | Λ | $\Delta \Lambda $ |
---|---|---|

$\Delta {\mathrm{CM}}_{\mathrm{h}}$ | $0.498\pm 0.026$ | $0.017\pm 0.058$ |

$\Delta {\sigma}_{\mathrm{h}}$ | $0.481\pm 0.050$ | $0.014\pm 0.100$ |

$n/{n}_{0}$ | $0.379\pm 0.024$ | $0.040\pm 0.056$ |

${\mathrm{FMS}}_{2\mathrm{D}}$ | $0.514\pm 0.037$ | $0.048\pm 0.085$ |

${\mathrm{FMS}}_{3\mathrm{D}}$ | $0.538\pm 0.023$ | $0.082\pm 0.063$ |

${\mathrm{PCC}}_{2\mathrm{D}}$ | $0.464\pm 0.023$ | $0.017\pm 0.051$ |

${\mathrm{PCC}}_{3\mathrm{D}}$ | $0.447\pm 0.022$ | $0.010\pm 0.049$ |

## 8. Summary

## Acknowledgments

## Conflicts of Interest

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Haszpra, T.
Time-Reversibility in Atmospheric Dispersion. *Atmosphere* **2016**, *7*, 11.
https://doi.org/10.3390/atmos7010011

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Haszpra T.
Time-Reversibility in Atmospheric Dispersion. *Atmosphere*. 2016; 7(1):11.
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2016. "Time-Reversibility in Atmospheric Dispersion" *Atmosphere* 7, no. 1: 11.
https://doi.org/10.3390/atmos7010011