# A Parametric Model of Elliptic Orbits for Annual Evolutions of Northern Hemisphere Stratospheric Polar Vortex and Their Interannual Variability

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Data

#### 2.2. Methods

_{0}) = 1 if x$\ge x$

_{0}, and otherwise H(x,x

_{0}) = 0. In (1) and (2) below, x = θ(λ,φ,σ,t) and x

_{0}= 400 K.

^{−1}), which can be obtained according to

## 3. Results

#### 3.1. Statistics of the SPV Indices

#### 3.2. The Parametric Elliptic Orbit Model

_{0}and Y

_{0}are the coordinates of the elliptic central point; a and b are the amplitudes; Θ is the phase angle in the range from 0 to $2\pi $; and a corresponds to the tilting angle of the elliptic obit with the x-axis ranging from −$\pi $ to $\pi $. The other form of (3) is the familiar elliptic equation, namely

_{0}, Y

_{0}, a, b and $\alpha $ with $\mathsf{\Theta}=\frac{2\pi t}{365}$, where t starts on 1 July as t = 1 and ends on June 30 as t = 365 (29 February in a leap year is excluded in our analysis). The information from “fit_ellipse” can be found at https://www.mathworks.com/matlabcentral/fileexchange/3215-fit_ellipse (accessed on 15 January 2023). To better capture the observed lead time information of MU with respect to M, we introduce an auxiliary equation when applying the “fit_ellipse” function such that the constructed time series of MU and M from the fitted elliptic orbit meet the same lead time of the observed MU with respect to the observed M. As illustrated in Figure 5, when the value of $\alpha $ is fixed, the lead-lag days of maximum correlation and the ratio b/a have a one-to-one correspondence relation. The climatological mean annual cycles of MU and M and their elliptic orbit are displayed in Figure 6. The close resemblance between the fitted orbit elliptic (dashed magenta) and the scattering plot of the observed MU and M (colored dots) indicates that an elliptic orbit is capable of capturing the annual evolutions of MU and M. In particular, the fitted MU also varies out of phase with the fitted M.

#### 3.3. Year-to-Year Variations of the Elliptic Orbits for the Annual Evolutions of M and MU

_{0}and smaller values of Y

_{0}than their counterparts for the climatological elliptic orbit are observed, and the general out-of-phase relationship between MU and M is also present in the year-to-year variation in their annual means. The generally larger values of a and b than their climatological counterparts indicate that the amplitudes of the seasonal cycles of individual years are stronger than those of the climatological annual cycle. The generally larger values of b/a and α than their climatological counterparts indicate that the lead days of MU with respect to M in individual years are longer than for their climatological counterpart.

## 4. Conclusions

_{0}, Y

_{0}, a, b, and α) of the corresponding yearly elliptic orbit. Because the timings of minimum MU and maximum M correspond closely to the timings of SSW events, one could in turn predict the timings of the high probability occurrence of SSW events in winters. Such a circulation condition during this timing would provide a favorable background for the break-up of the stratospheric polar vortex, which may yield a higher probability of the occurrence of SSW, which will be a topic of our future studies.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The 42-year climatological annual evolutions (red curves) and their interannual variability (blue shadings) in M, (

**a**) units: 10

^{16}kg and MU (

**b**) units: 10

^{12}kg s

^{−1}).

**Figure 2.**Annual evolutions (the ordinate) of (

**a**) M (units: 10

^{16}kg) and (

**b**) MU (units: 10

^{12}kg s

^{−1}) in the years from 1980 to 2020 (abscissa).

**Figure 3.**Time series of M (red curves, units: 10

^{16}kg) and MU (blue curves, units: 10

^{12}kg s

^{−1}) in four example years: (

**a**) 1980, (

**b**) 1990, (

**c**) 2000, (

**d**) 2010. The solid brown curves and light blue curves are 7-day running means of M (ordinate on the right-hand-side) and MU (ordinate on the left-hand-side), respectively; and dashed curves are original time series of M and MU, respectively. The solid magenta and blue curves in each panel (which are, respectively, identical in the four panels) are the 42-year climatology for M and MU, respectively.

**Figure 4.**Yearly time series of the lead days of MU with respect to M when their negative correlations reach maximum values in years from 1980 to 2021, derived from observational fields.

**Figure 5.**(

**a**) Orbits of two variables, X and Y, satisfy the idealized parametric ellipse orbit model, with the tilting angle of the ellipse with the horizontal axis ($\alpha $) equaling −0.25$\pi $. (

**b**) Lead days of X with respect to Y when their negative correlations reach maximum values as a function of the ratio of b to a and the tilting angle $\alpha $.

**Figure 6.**Scatter plot of the climatological mean M versus MU (colored dots) in the period 1980–2021 and its fitted ellipse orbit (dashed black curve).

**Figure 7.**Yearly time series of the parameters of the fitted elliptic orbit model for M and MU. (

**a**) The parameter X

_{0}, (

**b**) the parameter Y

_{0}, (

**c**) the parameter a, (

**d**) the parameter b, (

**e**) the parameter α, and (

**f**) the ratio of b to a. The horizontal dashed red lines are the values of the parameters for the fitted orbit of the climatological mean annual cycles of MU and M.

**Figure 8.**Scatter plot (color dots) of the M and MU in all years from 1980 to 2021 and the fitted elliptic orbit (black orbit).

**Figure 9.**As in Figure 2, but for the (

**a**) M and (

**b**) MU derived from the parametric model of ellipse orbit with the parameter b calculation scheme adjusted.

**Figure 10.**Scatter plot of the orbit-fitted extremes (red circles), and the observed extremes of maximum value of MU and minimum value of M in each winter season. (

**a**) The maximum of MU from orbit fitting (ordinate) versus the observation (abscissa), (

**b**) the same as (a) but for the minimum of M, (

**c**) the same as (a) but for the yearly difference between maximum and minimum values of MU, (

**d**) the same as (a) but for the difference between maximum and minimum values of M, (

**e**) the maximum of MU (abscissa) versus the minimum of M (ordinate) for observation (blue circles) and fitted orbits (red circles), and (

**f**) the same as (

**e**) but for the yearly differences between maximum and minimum values of MU and M.

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**MDPI and ACS Style**

Yu, Y.; Sun, J.; Secor, M.; Cai, M.; Luo, X.
A Parametric Model of Elliptic Orbits for Annual Evolutions of Northern Hemisphere Stratospheric Polar Vortex and Their Interannual Variability. *Atmosphere* **2023**, *14*, 870.
https://doi.org/10.3390/atmos14050870

**AMA Style**

Yu Y, Sun J, Secor M, Cai M, Luo X.
A Parametric Model of Elliptic Orbits for Annual Evolutions of Northern Hemisphere Stratospheric Polar Vortex and Their Interannual Variability. *Atmosphere*. 2023; 14(5):870.
https://doi.org/10.3390/atmos14050870

**Chicago/Turabian Style**

Yu, Yueyue, Jie Sun, Michael Secor, Ming Cai, and Xinyue Luo.
2023. "A Parametric Model of Elliptic Orbits for Annual Evolutions of Northern Hemisphere Stratospheric Polar Vortex and Their Interannual Variability" *Atmosphere* 14, no. 5: 870.
https://doi.org/10.3390/atmos14050870