# A Data-Driven Study of the Drivers of Stratospheric Circulation via Reduced Order Modeling and Data Assimilation

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## Abstract

**:**

## 1. Introduction

#### 1.1. General Background

- The interaction with waves propagating up from the troposphere;
- Differential radiative heating.

#### 1.2. Data Assimilation and Reduced-Order Models

#### 1.3. Our Main Focus

#### 1.4. Outline of the Paper

## 2. Materials and Methods

#### 2.1. Ruzmaikin Model

#### 2.2. ECMWF Data

#### 2.3. Data Assimilation

#### 2.3.1. Particle Filter

#### 2.3.2. ESMDA

- We start by sampling a large ensemble of realizations of the prior uncertain parameters, given their prescribed first-guess values and standard deviations;
- We then integrate the ensemble of model realizations forward in time to produce a prior ensemble prediction, which also characterizes the uncertainty;
- We compute the posterior ensemble of parameters by making use of the misfit between prediction and observations, and the correlations between the input parameters and the predicted measurements;
- Ultimately, we compute the posterior ensemble prediction by a forward ensemble integration. The posterior ensemble is then the “optimal” model prediction with the ensemble spread representing the uncertainty.

#### 2.3.3. Twin Model Analysis

## 3. Results

#### 3.1. Particle Filter

#### 3.1.1. Identical Twin Model Experiments

#### 3.1.2. ECMWF Data Assimilation

#### 3.1.3. Particle Filter Summary

#### 3.2. ESMDA Analysis

#### 3.2.1. Free Parameters

#### 3.2.2. Identical Twin Model Experiments

#### 3.2.3. Parameterized $\Lambda \left(t\right)$ and Free $h\left(t\right)$

#### 3.2.4. Free $\Lambda \left(t\right)$ and Constant h

#### 3.2.5. Free $\Lambda \left(t\right)$ and $h\left(t\right)$

#### 3.3. Analysis around SSW Events

## 4. Conclusions and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

^{−1}and $k=2/(acos\mathit{\pi}/3)=6.28\times {10}^{-7}$ m

^{−1}where a is the radius of the earth. For the boundary conditions, [6] assumes zero normal flow across the lateral boundaries, which requires that ${\mathit{\psi}}^{\prime}$ and ${v}^{\prime}$ vanish at the side boundary $y=0,L.$ The upper and bottom boundaries are chosen at ${z}_{B}=10$ km (tropopause) and ${z}_{T}=80$ km (mesopause), respectively. Further, Ref. [6] assumes that pertubation and zonal mean flow vanish at the upper boundary and specifies the lower boundary conditions by setting $\overline{u}={\overline{u}}_{B}\left(y\right)$ and ${\mathit{\psi}}^{\prime}={\mathit{\psi}}_{B}(y,t).$

Parameter | Value |
---|---|

${\mathit{\tau}}_{1}$ | 122.6276 |

r | 0.6286 |

s | 1.9638 |

$\mathit{\xi}$ | 1.7488 |

${\mathit{\delta}}_{w}$ | 70.8437 |

$\mathit{\zeta}$ | 240.5361 |

${U}_{R}$ | 0.4748 |

${\mathit{\tau}}_{2}$ | 30.3713 |

$\mathit{\eta}$ | $9.131\times {10}^{4}$ |

${\mathit{\delta}}_{\Lambda}$ | $4.9115\times {10}^{-4}$ |

${\Lambda}_{0}$ | 0.75 m/s/km |

$\mathit{\delta}{\Lambda}_{a}$ | 2.25 m/s/km |

$\mathit{\u03f5}$ | 0–0.3 |

## Appendix B

## Appendix C

**Figure A1.**Results from ESMDA fraternal twin experiments for $\Lambda \left(t\right)$ parameterized by ${\Lambda}_{0}=0.75$, ${\Lambda}_{a}=2.25$ m/s/km, and $\mathit{\u03f5}=0.3$ fixed and $h\left(t\right)$ free. (

**a**) RMSE (blue) and variance (orange) of model analysis $U\left(t\right)$ while (

**b**) shows the variance (blue) and mean (orange) of $h\left(t\right)$, each for varying values of ${\mathit{\tau}}_{h}$. (

**c**) Model output for variables normalized U (red), h (blue), and $\Lambda $ (black). (

**d**) Comparison of model estimated wind speeds (red) with the observed wind speeds (black). (

**e**) $\Lambda -U$ phase space of model output (blue) superimposed on the bifurcation diagram for $\Lambda $ (orange). (

**c**–

**e**) Model output is from the lowest RMSE experiment (${\mathit{\tau}}_{h}=91$).

**Figure A2.**Results from ESMDA fraternal twin experiments with $\Lambda \left(t\right)$ parameterized by ${\Lambda}_{0}$ and $\mathit{\u03f5}$ free, ${\Lambda}_{a}\ge 2.25$ m/s/km, and $h\left(t\right)$ free. (

**a**) RMSE (blue) and variance (orange) of model analysis $U\left(t\right)$ while (

**b**) shows the variance (blue) and mean (orange) of $h\left(t\right)$, each for varying values of ${\mathit{\tau}}_{h}$. (

**c**) Model output for variables normalized U (red), h (blue), and $\Lambda $ (black). (

**d**) Comparison of model estimated wind speeds (red) with the observed wind speeds (black). (

**e**) $\Lambda -U$ phase space of model output (blue) superimposed on the bifurcation diagram for $\Lambda $ (orange). (

**c**–

**e**) Model output is from the lowest RMSE experiment (${\mathit{\tau}}_{h}=547$).

## Appendix D

**Figure A3.**(

**a**–

**n**) $\Lambda $ (blue), U (orange), and h (red) around known SSW events with ${\mathit{\tau}}_{\mathit{\lambda}}=547$, ${\mathit{\tau}}_{h}=15$.

**Figure A4.**(

**a**–

**n**) $\Lambda $ (blue), U (orange), and h (red) around known SSW events with ${\mathit{\tau}}_{\mathit{\lambda}}=182$, ${\mathit{\tau}}_{h}=15$.

**Figure A5.**(

**a**–

**n**) $\Lambda $ (blue), U (orange), and h (red) around known SSW events with ${\mathit{\tau}}_{\mathit{\lambda}}=15$, ${\mathit{\tau}}_{h}=15$.

## References

- Muench, H.S. On the dynamics of the wintertime stratosphere circulation. J. Atmos. Sci.
**1965**, 22, 349–360. [Google Scholar] [CrossRef] - Geisler, J. A numerical model of the sudden stratospheric warming mechanism. J. Geophys. Res.
**1974**, 79, 4989–4999. [Google Scholar] [CrossRef] - Charney, J.G.; DeVore, J.G. Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci.
**1979**, 36, 1205–1216. [Google Scholar] [CrossRef] - Ambaum, M.H.P.; Hoskins, B.J. The NAO Troposphere–Stratosphere Connection. J. Clim.
**2002**, 15, 1969–1978. [Google Scholar] [CrossRef] - Finkel, J.; Abbot, D.S.; Weare, J. Path properties of atmospheric transitions: Illustration with a low-order sudden stratospheric warming model. J. Atmos. Sci.
**2020**, 77, 2327–2347. [Google Scholar] [CrossRef] - Holton, J.R.; Mass, C. Stratospheric vacillation cycles. J. Atmos. Sci.
**1976**, 33, 2218–2225. [Google Scholar] [CrossRef] - McIntyre, M.E. How well do we understand the dynamics of stratospheric warmings? J. Meteorol. Soc. Japan. Ser. II
**1982**, 60, 37–65. [Google Scholar] [CrossRef] - Scott, R.; Polvani, L.M. Internal variability of the winter stratosphere. Part I: Time-independent forcing. J. Atmos. Sci.
**2006**, 63, 2758–2776. [Google Scholar] [CrossRef] - Stan, C.; Straus, D.M. Stratospheric predictability and sudden stratospheric warming events. J. Geophys. Res. Atmos.
**2009**, 114. [Google Scholar] [CrossRef] - Jeppesen, J. Fact Sheet: Reanalysis. 2020. Available online: https://www.ecmwf.int/en/about/media-centre/focus/2020/fact-sheet-reanalysis (accessed on 31 July 2021).
- Kantas, N.; Doucet, A.; Singh, S.S.; Maciejowski, J.; Chopin, N. On particle methods for parameter estimation in state-space models. Stat. Sci.
**2015**, 30, 328–351. [Google Scholar] [CrossRef] - Gordon, N.J.; Salmond, D.J.; Smith, A.F. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. In Proceedings of the IEE Proceedings F-Radar and Signal Processing; IET: Stevenage, UK, 1993; Volume 140, pp. 107–113. [Google Scholar]
- Asch, M.; Bocquet, M.; Nodet, M. Data Assimilation: Methods, Algorithms, and Applications; SIAM, Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2016. [Google Scholar]
- Bocquet, M.; Sakov, P. An iterative ensemble Kalman smoother. Q. J. R. Meteorol. Soc.
**2013**, 140, 1521–1535. [Google Scholar] [CrossRef] - Carrassi, A.; Bocquet, M.; Bertino, L.; Evensen, G. Data Assimilation in the Geosciences—An overview on methods, issues and perspectives. arXiv
**2018**, arXiv:1709.02798. [Google Scholar] [CrossRef] - Skjervheim, J.; Evensen, G. An ensemble smoother for assisted history matching. In Proceedings of the SPE Reservoir Simulation Symposium, Woodlands, TX, USA, 21–23 February 2011; SPE 141929. SPE: Richardson, TX, USA, 2011. [Google Scholar] [CrossRef]
- Ruzmaikin, A.; Lawrence, J.; Cadavid, C. A simple model of stratospheric dynamics including solar variability. J. Clim.
**2003**, 16, 1593–1600. [Google Scholar] [CrossRef] - Andrews, D.G.; Holton, J.R.; Leovy, C.B. Middle Atmosphere Dynamics; Number 40; Academic Press: Cambridge, MA, USA, 1987. [Google Scholar]
- Eichelberger, S.J.; Holton, J.R. A mechanistic model of the northern annular mode. J. Geophys. Res. Atmos.
**2002**, 107, ACL 10-1–ACL 10-12. [Google Scholar] [CrossRef] - Wakata, Y.; Uryu, M. Stratospheric multiple equilibria and seasonal variations. J. Meteorol. Soc. Japan. Ser. II
**1987**, 65, 27–42. [Google Scholar] [CrossRef] - Yoden, S. An illustrative model of seasonal and interannual variations of the stratospheric circulation. J. Atmos. Sci.
**1990**, 47, 1845–1853. [Google Scholar] [CrossRef] - Sanchez, K. Understanding the Stratospheric Polar Vortex: A Parameter Sensitivity Analysis on a Simple Model of Stratospheric Dynamics. Ph.D. Thesis, Sam Houston State University, Huntsville, TX, USA, 2020. [Google Scholar]
- Berrisford, P.; Kållberg, P.; Kobayashi, S.; Dee, D.; Uppala, S.; Simmons, A.; Poli, P.; Sato, H. Atmospheric conservation properties in ERA-Interim. Q. J. R. Meteorol. Soc.
**2011**, 137, 1381–1399. [Google Scholar] [CrossRef] - Wu, Z. Data Assimilation and Uncertainty Quantification with Reduced-order Models. Ph.D. Thesis, Arizona State University, Tempe, AZ, USA, 2021. [Google Scholar]
- Smith, A.F.; Gelfand, A.E. Bayesian statistics without tears: A sampling–resampling perspective. Am. Stat.
**1992**, 46, 84–88. [Google Scholar] - Marín, D.A.A. Particle Filter Tutorial. 2012. Available online: https://www.mathworks.com/matlabcentral/fileexchange/35468-particle-filter-tutorial (accessed on 2 April 2021).
- Pfister, R.; Schwarz, K.A.; Janczyk, M.; Dale, R.; Freeman, J. Good things peak in pairs: A note on the bimodality coefficient. Front. Psychol.
**2013**, 4, 700. [Google Scholar] [CrossRef] - Freeman, J.B.; Dale, R. Assessing bimodality to detect the presence of a dual cognitive process. Behav. Res. Methods
**2013**, 45, 83–97. [Google Scholar] [CrossRef] - Emerick, A.A.; Reynolds, A.C. Ensemble smoother with multiple data assimilation. Comput. Geosci.
**2013**, 55, 3–15. [Google Scholar] [CrossRef] - Fleurantin, E.; Sampson, C.; Maes, D.P.; Bennett, J.; Fernandes-Nunez, T.; Marx, S.; Evensen, G. A study of disproportionately affected populations by race/ethnicity during the SARS-CoV-2 pandemic using multi-population SEIR modeling and ensemble data assimilation. Found. Data Sci.
**2021**, 3, 479–541. [Google Scholar] [CrossRef] - Evensen, G. EnKF_seir. 2021. Available online: https://github.com/geirev/EnKF_seir (accessed on 30 January 2021).
- Lawson, L.M.; Spitz, Y.H.; Hofmann, E.E.; Long, R.B. A data assimilation technique applied to a predator-prey model. Bull. Math. Biol.
**1995**, 57, 593–617. [Google Scholar] [CrossRef] - Houtekamer, P.L.; Mitchell, H.L. Data assimilation using an ensemble Kalman filter technique. Mon. Weather Rev.
**1998**, 126, 796–811. [Google Scholar] [CrossRef] - Browne, P.; Van Leeuwen, P. Twin experiments with the equivalent weights particle filter and HadCM3. Q. J. R. Meteorol. Soc.
**2015**, 141, 3399–3414. [Google Scholar] [CrossRef] - Evensen, G. The Ensemble Kalman Filter: Theoretical formulation and practical implementation. Ocean Dyn.
**2003**, 53, 343–367. [Google Scholar] [CrossRef] - Evensen, G. Sampling strategies and square root analysis schemes for the EnKF. Ocean Dyn.
**2004**, 54, 539–560. [Google Scholar] [CrossRef] - Yoden, S. Bifurcation properties of a stratospheric vacillation model. J. Atmos. Sci.
**1987**, 44, 1723–1733. [Google Scholar] [CrossRef] - Butler, A.H. Table of Major Mid-Winter SSWs in Reanalyses Products; NOAA Chemical Science Laboratory: Boulder, CO, USA, 2020. Available online: https://csl.noaa.gov/groups/csl8/sswcompendium/majorevents.html (accessed on 6 July 2023).
- Butler, A.H.; Seidel, D.J.; Hardiman, S.C.; Butchart, N.; Birner, T.; Match, A. Defining Sudden Stratospheric Warmings. Bull. Am. Meteorol. Soc.
**2015**, 96, 1913–1928. [Google Scholar] [CrossRef] - Charlton, A.J.; Polvani, L.M. A New Look at Stratospheric Sudden Warmings. Part I: Climatology and Modeling Benchmarks. J. Clim.
**2007**, 20, 449–469. [Google Scholar] [CrossRef] - Zuev, V.V.; Savelieva, E. Arctic polar vortex dynamics during winter 2006/2007. Polar Sci.
**2020**, 25, 100532. [Google Scholar] [CrossRef] - Manabe, S.; Wetherald, R.T. Thermal Equilibrium of the Atmosphere with a Given Distribution of Relative Humidity. J. Atmos. Sci.
**1967**, 24, 241–259. [Google Scholar] [CrossRef] - Santer, B.D.; Po-Chedley, S.; Zhao, L.; Zou, C.Z.; Fu, Q.; Solomon, S.; Thompson, D.W.J.; Mears, C.; Taylor, K.E. Exceptional stratospheric contribution to human fingerprints on atmospheric temperature. Proc. Natl. Acad. Sci. USA
**2023**, 120, e2300758120. [Google Scholar] [CrossRef] - Holton, J.R.; Dunkerton, T. On the role of wave transience and dissipation in stratospheric mean flow vacillations. J. Atmos. Sci.
**1978**, 35, 740–744. [Google Scholar] [CrossRef] - Keevil, J. ODE4 Gives More Accurate Results than ODE45, ODE23, ODE23s. 2016. Available online: https://www.mathworks.com/matlabcentral/fileexchange/59044-ode4-gives-more-accurate-results-than-ode45-ode23-ode23s (accessed on 13 May 2021).

**Figure 2.**(

**a**–

**d**) RMSEs for combinations of assimilation period and observation error when the state vector and control parameters h and $\Lambda $ are estimated simultaneously using synthetic data from the Ruzmaikin model. (

**e**) Bimodality as measured by the percentage of timepoints with sufficiently large Sarle statistic. (

**f**) An example of a bimodal ensemble distribution with a Sarle statistic value of $\mathrm{BC}=0.728$, with ${\mathit{\sigma}}_{\mathrm{obs}}^{2}=0.1$ and assimilation period of 3 weeks.

**Figure 3.**(

**a**) RMSE for mean zonal wind using the particle filter with ECMWF reanalysis data. (

**b**–

**d**) Estimates for h and the coefficients of $\Lambda $ obtained by time-averaging the ensemble mean over the last year of analysis.

**Figure 5.**Example model outputs from the ESMDA identical twin experiments, each with ${\mathit{\tau}}_{h}=91$. (

**a**–

**c**) show the model analysis $\Lambda \left(t\right)$ (red) versus the known, true $\Lambda \left(t\right)$ (black), while (

**d**–

**f**) show these curves for mean zonal wind speed $U\left(t\right)$. (

**g**–

**i**) Model output (blue) translated into $\Lambda -U$ phase space superimposed on the bifurcation diagram for $\Lambda $. Note that the bifurcation diagrams only showcases the stable equilibrium branches (orange), and this is repeated in subsequent figures. Decorrelation lengths for $\Lambda $ are set at (

**a**,

**d**,

**g**): ${\mathit{\tau}}_{\mathit{\lambda}}=547$, (

**b**,

**e**,

**h**): ${\mathit{\tau}}_{\mathit{\lambda}}=182$, and (

**c**,

**f**,

**i**): ${\mathit{\tau}}_{\mathit{\lambda}}=15$.

**Figure 6.**Results from ESMDA fraternal twin experiments for $\Lambda \left(t\right)$ parameterized by estimated ${\Lambda}_{0}$, ${\Lambda}_{a}$, and $\mathit{\u03f5}$ and $h\left(t\right)$ free. (

**a**) RMSE (blue) and variance (orange) of model analysis $U\left(t\right)$, while (

**b**) shows variance (blue) and mean (orange) of $h\left(t\right)$, each for varying values of ${\mathit{\tau}}_{h}$. (

**c**) Model output for normalized variables $U,$$h,$ and $\Lambda $. (

**d**) Comparison of estimated wind speeds (red) and observed wind speeds (black). (

**e**) $\Lambda -U$ phase space of model output (blue) superimposed on the bifurcation diagram for $\Lambda $ (orange). (

**c**–

**e**) Model output is from the lowest RMSE experiment (${\mathit{\tau}}_{h}=15$).

**Figure 7.**Results from ESMDA fraternal twin experiments for $\Lambda \left(t\right)$ free and $h\left(t\right)$ constant. (

**a**) RMSE (blue) and variance (orange) of model analysis $U\left(t\right)$, while (

**b**) shows the variance (blue) and mean (orange) of $\Lambda \left(t\right)$, each for varying values of ${\mathit{\tau}}_{\mathit{\lambda}}$.

**Figure 8.**Example model results for $\Lambda \left(t\right)$ free and h constant. (

**a**–

**c**) Model output for normalized variables U (red) and $\Lambda $ (black). (

**d**–

**f**) Comparison of model estimated wind speeds (red) with the observed wind speeds (black). (

**g**–

**i**) $\Lambda -U$ phase space of model output (blue) superimposed on the bifurcation diagram for $\Lambda $ (orange). Decorrelation lengths for $\Lambda $ are set at (

**a**,

**d**,

**g**): ${\mathit{\tau}}_{\mathit{\lambda}}=547$, (

**b**,

**e**,

**h**): ${\mathit{\tau}}_{\mathit{\lambda}}=182$, and (

**c**,

**f**,

**i**): ${\mathit{\tau}}_{\mathit{\lambda}}=30$.

**Figure 9.**Summary statistics for ESMDA fraternal twin experiments with free $\Lambda \left(t\right)$ and $h\left(t\right)$ and varying decorrelation lengths.

**Figure 10.**Example model results with both $\Lambda \left(t\right)$ and $h\left(t\right)$ free and ${\mathit{\tau}}_{h}=365$. (

**a**–

**c**) Model output for normalized variables U (red), h (blue), and $\Lambda $ (black). (

**d**–

**f**) Comparison of model estimated wind speeds (red) with the observed wind speeds (black). (

**g**–

**i**) $\Lambda -U$ phase space of model output superimposed on the bifurcation diagram for $\Lambda .$ Decorrelation lengths for $\Lambda $ are set at (

**a**,

**d**,

**g**): ${\mathit{\tau}}_{\mathit{\lambda}}=547$, (

**b**,

**e**,

**h**): ${\mathit{\tau}}_{\mathit{\lambda}}=182$, and (

**c**,

**f**,

**i**): ${\mathit{\tau}}_{\mathit{\lambda}}=15$.

**Figure 11.**Examples of model output around known SSW events for U (orange), $\Lambda $ (blue), and h (red), each with ${\mathit{\tau}}_{h}=365$. Decorrelation lengths for $\Lambda $ are set at (

**a**–

**c**): ${\mathit{\tau}}_{\mathit{\lambda}}=547$, (

**d**–

**f**): ${\mathit{\tau}}_{\mathit{\lambda}}=182$, and (

**g**–

**i**): ${\mathit{\tau}}_{\mathit{\lambda}}=15$.

**Figure 12.**Examples of model output around known SSW events for U (orange), $\Lambda $ (blue), and h (red), each with ${\mathit{\tau}}_{h}=15$. Decorrelation lengths for $\Lambda $ are set at (

**a**–

**c**): ${\mathit{\tau}}_{\mathit{\lambda}}=547$, (

**d**–

**f**): ${\mathit{\tau}}_{\mathit{\lambda}}=182$, and (

**g**–

**i**): ${\mathit{\tau}}_{\mathit{\lambda}}=15$.

**Figure 13.**Mean (blue) and variance (orange) of (

**a**–

**c**): $\Lambda \left(t\right)$ and (

**d**–

**f**): $U\left(t\right)$ over several different moving windows, 1 year, 2 years, and 5 years.

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## Share and Cite

**MDPI and ACS Style**

Sherman, J.; Sampson, C.; Fleurantin, E.; Wu, Z.; Jones, C.K.R.T.
A Data-Driven Study of the Drivers of Stratospheric Circulation via Reduced Order Modeling and Data Assimilation. *Meteorology* **2024**, *3*, 1-35.
https://doi.org/10.3390/meteorology3010001

**AMA Style**

Sherman J, Sampson C, Fleurantin E, Wu Z, Jones CKRT.
A Data-Driven Study of the Drivers of Stratospheric Circulation via Reduced Order Modeling and Data Assimilation. *Meteorology*. 2024; 3(1):1-35.
https://doi.org/10.3390/meteorology3010001

**Chicago/Turabian Style**

Sherman, Julie, Christian Sampson, Emmanuel Fleurantin, Zhimin Wu, and Christopher K. R. T. Jones.
2024. "A Data-Driven Study of the Drivers of Stratospheric Circulation via Reduced Order Modeling and Data Assimilation" *Meteorology* 3, no. 1: 1-35.
https://doi.org/10.3390/meteorology3010001