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Citation: Sherman, J.; Sampson, C.;

Fleurantin, E.; Wu, Z.; and Jones,

C.K.R.T.A data driven study of the

drivers of stratospheric circulation via

reduced order modeling and data

assimilation. Meteorology 2023,1, 1–28.

https://doi.org/

Received:

Revised:

Accepted:

Published:

Copyright: © 2023 by the authors.

Submitted to Meteorology for

possible open access publication

under the terms and conditions

of the Creative Commons Attri-

bution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

Article

A data driven study of the drivers of stratospheric circulation via

reduced order modeling and data assimilation

Julie Sherman 1*, Christian Sampson 2, Emmanuel Fleurantin 3, Zhimin Wu 4, and Christopher K.R.T. Jones 5

1University of Utah; sherman@math.utah.edu

2Joint Center for Satellite Data Assimilation; christian.sampson@gmail.com

3George Mason University, University of North Carolina at Chapel Hill; eﬂeuran@gmu.edu

4Arizona State University; zhiminwu@asu.edu

5

George Mason University, University of North Carolina at Chapel Hill, University of Reading; ckrtj@renci.org

*Correspondence: sherman@math.utah.edu

Abstract: Stratospheric dynamics are strongly affected by the absorption/emission of radiation in

1

the Earth’s atmosphere and Rossby waves that propagate upward from the troposphere, perturbing

2

zonal ﬂow. Reduced order models of stratospheric wave-zonal interactions, which parameterize these

3

effects, have been used to study interannual variability in stratospheric zonal winds and sudden

4

stratospheric warming (SSW) events. These models are most sensitive to two main parameters:

Λ

,

5

forcing the mean radiative zonal wind gradient, and

h

, a perturbation parameter representing the

6

effect of Rossby waves. We take one such reduced order model (Ruzmaikin et al., 2003) with 20 years

7

of ECMWF atmospheric reanalysis data and estimate

Λ

and

h

using both a particle ﬁlter and an

8

ensemble smoother, investigating which parameter properties are needed to match the reanalysis.

9

Allowing for sufﬁcient variability in

Λ

, we can ﬁt parameters such that the model output closely

10

matches the reanalysis data and is largely consistent with the dynamics of the reduced-order model.

11

Furthermore, our analysis shows physical signatures in its parameter estimates around known SSW

12

events. This work provides a data-driven examination of these important parameters representing

13

fundamental stratospheric processes through the lens and tractability of a reduced order model,

14

shown to be physically representative of the relevant atmospheric dynamics. 15

Keywords: particle ﬁlter; ensemble smoother; sudden stratospheric warming; bistability; reanalysis

16

data; dynamical systems; synoptic-scale meteorology 17

1. Introduction 18

1.1. General Background 19

Stratospheric polar circulation in the northern hemisphere, often referred to as the

20

polar vortex, is a counterclockwise cyclonic circulation resulting from the strong tempera-

21

ture difference between the polar and subtropical regions which forms during the autumn

22

and winter as solar radiation in the polar region diminishes. This cyclonic circulation

23

extends from the surface to the upper part of the stratosphere often referred to separately

24

as the tropospheric (near the surface) and stratospheric (near the top of the stratosphere)

25

polar vortex. Changes in the stratospheric polar vortex can have strong effects on the

26

tropospheric vortex and thus it plays a signiﬁcant role in winter weather conditions in the

27

northern hemisphere. 28

For decades, models of stratospheric circulation have sought to explain and predict

29

variability in wintertime polar vortex dynamics via mechanistic and/or phenomenological

30

processes [

1

–

5

]. A well-studied method of stratospheric modeling uses a quasi-geostrophic

31

β−

plane channel model to study wave-mean zonal ﬂow interactions. An early model of this

32

type by Holton and Mass [

6

] and its many extensions depend highly on two fundamental

33

processes: 34

1. The interaction with waves propagating up from the troposphere and 35

Version September 30, 2023 submitted to Meteorology https://www.mdpi.com/journal/meteorology

Version September 30, 2023 submitted to Meteorology 2

2. differential radiative heating. 36

Furthermore, such models have been shown to exhibit mathematical properties consistent

37

with atmospheric realizations including multiple stable or metastable equilibria associated

38

with atmospheric blocking in the presence of external forcing [3,7–10]. 39

The two physical factors above are known to radically inﬂuence the polar vortex and

40

are included as important control parameters in these models. First, the weakening of the 41

polar vortex can be caused by large amplitude Rossby waves, also known as planetary

42

waves, which propagate upward from the ground and interfere with the zonal wind in

43

the stratosphere. Rossby waves are caused by topography as well as land-sea contrasts

44

and can be hard to measure directly. Another factor that inﬂuences the polar vortex is

45

the vertical gradient of radiative zonal wind. This is associated with solar forcing and is

46

largest when there is high differential heating. Indeed, it is the changes in the differential

47

heating throughout the seasons that lead to the annual formation and breakdown of the

48

polar vortex in the fall and spring, respectively. 49

A sudden stratospheric warming (SSW) event can also lead to a breakdown in the

50

polar vortex which can cause severe weather at lower latitudes. An SSW event can be

51

generically categorized by a rapid warming of the stratosphere near the pole which leads

52

to a decrease in differential heating weakening and even collapsing the cyclonic circulation

53

of the polar vortex. Two theories have been proposed to explain the occurrence of SSWs

54

that correspond with the control parameters of the reduced order models. First, enhanced

55

planetary waves are known to transfer momentum and heat from the troposphere to the

56

stratosphere, which alters the large-scale circulation [

11

,

12

]. Alternatively, it is understood

57

that the vertical shear in the stratosphere is key to controlling upward propagation of waves

58

through the lower boundary. Therefore, for smaller vertical temperature gradients, waves

59

are more disruptive to zonal stratospheric ﬂow [12,13]. 60

Reconstructions of global weather patterns are routinely done by marrying available

61

historical data to large-scale global circulation models through data assimilation (DA).

62

Having a time record of global observations is key for advancement, however, they are

63

irregular in space, time, and quality. By combining data and model, a complete picture of

64

a global weather pattern time series can be formed. An example of such a reanalysis can

65

be found through the European Center for Medium-Range Weather Forecasts (ECMWF)

66

[

14

]. These data sets can aid scientists working to understand geophysical processes and

67

changing climate conditions and develop new tools to improve predictions. 68

While large reanalysis data sets provide a plethora of information about the earth

69

system they are extremely complex and lack the tractability of reduced order models which

70

highlight the most important physical processes at play and can provide strong concep-

71

tual understanding often leading to new ideas. By considering only the most inﬂuential

72

processes, one can build a simpliﬁed model of stratospheric dynamics that maintains a

73

mechanistic understanding of the system while ignoring or averaging components less

74

relevant to large-scale and long-term behavior. In this work, we aim to leverage the complex

75

information available in a large reanalysis data set via DA to provide estimates and assess

76

the dynamics of important, physically-based parameters in the context of a reduced order

77

model. 78

1.2. Data Assimilation and Reduced-Order Models 79

Data assimilation has extremely useful applications in reduced order modeling. In

80

particular, it can give a sense of suitability of the simpliﬁed model to observed data as well

81

as estimate important parameters representing physical characteristics. DA also allows for

82

simultaneous estimation of model parameters via a number of methods (for an overview

83

see [

15

]). Assimilating reanalysis data with a reduced order model therefore provides a

84

convenient way to assess the model and estimate the values and relevance of essential

85

parameters controlling the system. 86

There are a great number of DA algorithms adapted to address a variety of situations.

87

One of the most general is the particle ﬁlter, also known as the bootstrap ﬁlter. This is an

88

Version September 30, 2023 submitted to Meteorology 3

ensemble-based, recursive algorithm that uses resampling and Bayes’ Theorem to derive

89

an empirical distribution which is asymptotically equivalent to the true distribution. This

90

algorithm is more general than the better known Kalman ﬁlter as it does not assume

91

linearity of the system nor normality of distributions [16]. 92

We also explore the use of an Ensemble Smoother with Multiple Data Assimilation

93

(ESMDA), updating the parameters that result in bias in the solution. This method allows

94

for the estimation of time dependent parameters since ﬁtting their values at a given time

95

can be adjusted to predict data at later times. Consequently, we are updating the parameters

96

in the past to remove the bias in the solution before it happens. Some examples of iterative

97

ensemble smoothers can be found in [

17

,

18

], in the geosciences in [

19

], or petroleum

98

reservoir modeling in [20]. 99

1.3. Our Main Focus 100

In this work, multiple data assimilation schemes were applied to a reduced order

101

model of the polar vortex published by Ruzmaikin et al. in 2003 [

10

]. A highly reduced

102

version of the original Holton and Mass model [

6

], this model is a nonlinear system of

103

ordinary differential equations which exhibits bistability for certain parameter ranges [

10

].

104

Its dynamics are closely tied to parameters related to radiative forcing due to differential

105

heating,

Λ

, and interactions with large-amplitude Rossby waves propagating upwards

106

from the troposphere,

h

. The bistability of this model is relevant to meteorological phe-

107

nomena including multiple stable states associated with atmospheric blocking events. This

108

motivates the use of the particle ﬁlter, as we expect bimodal ensemble distributions. How-

109

ever, the particle ﬁlter ultimately fails to accurately describe the reanalysis data, so we turn

110

to ESMDA, as it allows for greater degrees of freedom in parameter estimation. We ﬁnd

111

that by allowing

Λ(t)

to be un-parameterized, we can accurately reproduce reanalysis wind

112

speeds. Thus, we are able to provide insights into patterns and trends in physically-based

113

parameters using the simplicity of a reduced-order model while ensuring its applicability

114

through assimilating reanalysis data. 115

1.4. Outline of the Paper 116

The outline of the paper is as follows. In Section 2we discuss the materials and

117

methods. We present the Ruzmaikin model in 2.1, the ECMWF data used in 2.2, and our

118

data assimilation analysis methods in 2.3. In Section 3, we discuss our results: ﬁrst using

119

the particle ﬁlter in 3.1, then with ESMDA in 3.2. With the particle ﬁlter, we consider

120

hyperparemater estimation and examine ECMWF data assimilation. With ESMDA, we

121

deliberate on the parameters that we are estimating, the calibration of our runs with

122

twin experiments, and different scenarios for estimating control parameters

Λ(t)

and

h(t)

.

123

Then, in Section 3.3 we explore our ESMDA analysis results in the context of historical

124

atmospheric conditions, including around SSW events and trends over the 20-year period.

125

Finally, Section 4is dedicated to discussing our approach and concluding remarks. 126

2. Materials and Methods 127

2.1. Ruzmaikin Model 128

Considering the two driving forces highlighted above, Ruzmaikin et al. (2003) de-

129

veloped a simple dynamical model composed of three ordinary differential equations

130

(ODEs) that describe an atmospheric system localized as one point in the stratosphere. The

131

“Ruzmaikin model” is a highly truncated version of the Holton and Mass 1976 model (the

132

“HM76” model) of stratospheric wave-zonal ﬂow interactions. It is obtained by consid-

133

ering only one longitudinal and one latitudinal mode of the HM76 model and ﬁxing the

134

vertical level to 25 km log-pressure height using ﬁnite differences. Although such a one-

135

dimensional model cannot realistically describe the complicated stratospheric dynamics,

136

it captures the essential mechanism of interactions between planetary waves, radiative

137

forcing, and the zonal wind. 138

Version September 30, 2023 submitted to Meteorology 4

The Ruzmaikin model appears in the ﬁnal form of three ODEs with state variables,

X

,

Y

, and

U

, where

X

and

Y

represent the real and imaginary parts of the streamfunction,

respectively, and

U

represents mean zonal wind velocity. The system of ODE’s is given by

˙

X=−X/τ1−rY +sUY −ξh+δw˙

h(1)

˙

Y=−Y/τ1+rX −sUX +ζhU (2)

˙

U=−(U−UR)/τ2−ηhY −δΛ˙

Λ. (3)

Two control parameters are used in HM76 as well as the Ruzmaikin model, and all

139

other parameters are ﬁxed at their typical atmospheric values (Table A1). The ﬁrst control

140

parameter is the vertical gradient of the mean radiative zonal wind,

Λ(t) = dUR/dz

,

141

where it is assumed that

UR(z

,

t) = UR(

0,

t) + Λ(t)z

.

Λ(t)

is a time-dependent parameter

142

accounting both the seasonal variability and the 11-year solar cycle variability of solar

143

radiation. Speciﬁcally, Λis given the form 144

Λ(t) = Λ0+Λasin2πt

1 year +ϵΛ0sin2πt

11 year . (4)

The other control parameter characterizes the initial planetary wave amplitude and

145

is denoted by

h

. Speciﬁcally, it is equivalent to the perturbation at the ground level,

146

related to the wave streamfunction,

Ψ

, by

h(t) = Ψ(

0,

t)f0/g

. While several works have

147

explored various time or spatially-dependent parameterizations of

h

in similar models

148

[

6

,

7

,

9

], Ruzmaikin et al. ﬁxes

h

as constant. Sensitivity analysis conﬁrms that the model is

149

most sensitive to the two parameters hand Λ[21]. 150

Further analysis of the inﬂuence of

Λ

and

h

(considered independently as constants)

151

on equilibrium solutions show the existence of pitchfork bifurcations leading to bistability

152

[

10

, Fig. 2, Fig. 3]. For instance, with relatively small values of

h

corresponding to low

153

amplitude Rossby waves, there is a single equilibrium of relatively large mean zonal wind

154

(

Ue≈

35 m/s). For very large values of

h

, corresponding to high amplitude Rossby waves,

155

there is a single equilibrium of low mean zonal wind (

Ue≈

21 m/s). However, between

156

these two extremes there is an area of bistability, for which both strong and weak polar

157

vortices are achievable for the same value of

h

. Similarly, varying a constant

Λ

also leads

158

to a bifurcation of equilibrium values of zonal wind [

10

, Fig. 2]. Note again the region of

159

bistability for Λ⪆0.75 m/s/km when h=68m is ﬁxed. 160

2.2. ECMWF Data 161

Twenty years (1999 to 2018) of zonal wind data are obtained from European Center

162

for Medium-Range Weather Forecasts (ECMWF) Reanalysis - Interim (or “ERA-Interim”), 163

a global atmospheric reanalysis available from 1979. The reanalysis is based on a 2006

164

release of the ECWMF’s Integrated Forecast System (IFS). The data assimilation system of

165

ERA-Interim uses a 4-dimensional variational analysis (4D-Var) with a 12-hour analysis

166

window. More details of the ERA-Interim archive can be found in [

22

] and [

23

, Section 6.2].

167

The zonal wind of the Ruzmaikin model (variable

U

in equations (1)-(3)) is equivalent

168

to “

U

component of wind” provided by ECMWF Reanalysis - Interim archive. This data

169

is available at the 1

◦×

1

◦

horizontal, 10 mb vertical (in the upper stratosphere), and four

170

times daily resolution. Thus, we average according to the assumptions of the Ruzmaikin

171

model: the vertical level is ﬁxed at 25 km log-pressure height, and the latitudinal channel

172

is centered at 60

◦

N. First, daily mean zonal wind data is obtained by averaging over the

173

four given wind data values per day. Thus, the processed data set provides zonal wind

174

“observations” as daily averages of the wind from 1 January 1999 to 31 December 2018.

175

Next, daily averaged data from the pressure levels of 20 mb and 30 mb are interpolated to

176

25 km log-pressure height by linear approximation in the log-pressure vertical coordinate.

177

Finally, as the Ruzmaikin model is conﬁned to a latitudinal channel centered at 60

◦

N

178

with a meridional extent of 60

◦

latitude, daily means of zonal wind interpolated to 25km

179

log-pressure height are then averaged over a latitudinal window centered at 60

◦

N with a

180

Version September 30, 2023 submitted to Meteorology 5

(a)

(b)

Figure 1. Averaged ECMWF ERA-Interim data.

meridional extent of 20

◦

latitude. Note that we also tested larger meridional window sizes,

181

but these results are excluded, as the greater amount of averaging generally produced

182

lower wind speeds and muted winter “peaks,” as seen in Fig. 1b.183

We note that there is evidence of bistability in the ECMWF data, evidenced in Fig. 1.

184

Indeed, in Fig. 1a, we see examples of two dates with representative behaviors of the

185

polar vortex. On one hand, the jet may have high winds that are centered around the

186

north pole. However, when the jet is destabilized, it rotates at lower speeds, and can be

187

off-centered. These multiple winter states are also evident in the full-averaged data in

188

Fig. 1b. In particular with smaller meridional averaging windows, winter peak winds are

189

either high (

∼

35

−

45 m/s) or low (

∼

20

−

30 m/s). Thus, it is our purpose to understand

190

how the bistability of the reduced-order model relates to the bistability of the ERA-Interim

191

data via data assimilation. 192

Version September 30, 2023 submitted to Meteorology 6

2.3. Data Assimilation 193

2.3.1. Particle Filter 194

The ﬁrst DA method explored is the particle ﬁlter. This method is desirable because it

195

allows for nonlinear dynamics and requires no distributional assumptions. In particular,

196

due to the bistability of the system, we expect an ensemble of state variables to exhibit

197

bimodality, thereby violating the normality assumption of the more common Kalman ﬁlter.

198

A detailed description of the mathematical and algorithmic framework behind the

199

particle ﬁlter as well as its implementation in this context can be found in Appendix

200

B. Essentially, the particle ﬁlter uses recursion and Bayes’s rule (5) to approximate the

201

distribution of the state vector at discrete time

k

, denoted

xk

, given the set of all observations,

202

yi, up to and including time k,203

p(xk|y1, . . . , yk) = p(yk|xk)p(xk|y1, . . . , yk−1)

p(yk|y1, . . . , yk−1). (5)

Lacking an analytical solution in the general case, one can instead use an iterative

204

process of simulation and resampling to approximate the desired distribution. Indeed, each

205

time an observation

yk

, is obtained, if an ensemble (of size

nens

) of forecasted state vectors

206

{x∗

k(i):i=1, . . . , nens}is resampled according to the normalized probabilities 207

qi=p(yk|x∗

k(i))

∑nens

j=1p(yk|x∗

k(j)) , (6)

then the updated ensemble {xk(i):i=1, . . . , nens }is distributed as p(xk|y1, . . . , yk)[24]. 208

Parameter estimation is easily realized by appending the state vector

xk

of the dynam-

209

ical model with the parameters of interest. In this case, 210

xk=X Y U h Λ0ΛaT, (7)

where

Λ0

and

Λa

are coefﬁcients of the prescribed form for

Λ(t)

in Eqn. (4). The observed

211

variable, mean zonal wind speed, is

yk=U(k)

. Note that we also tested the particle ﬁlter

212

with

xk=X Y U hT

, and

xk=XYUΛ0ΛaT

, but generally the ﬁts were

213

inferior, and the results are omitted. 214

This algorithm was implemented in MATLAB via adaptations to a publicly available

215

particle ﬁlter tutorial [25]. 216

2.3.2. ESMDA 217

The next method of DA used in this paper is ESMDA. Ensemble smoother techniques

218

can be derived by assuming a perfect forward model. 219

y=g(x)(8)

In general,

x

is the realization of model parameters, and

y

consists of the uniquely

220

predicted measurements. We want to ﬁnd the set of model parameters

x

which produce

221

the observed data. 222

Assume that the observations dare perturbed stochastically from the truth 223

d←y+e, (9)

where erepresent errors from our model. This can be formulated as a Bayesian problem 224

f(x|d)∝f(d|g(x)) f(x). (10)

Version September 30, 2023 submitted to Meteorology 7

This deﬁnes the so-called smoothing problem. Our current approach is to use ensemble

225

methods to approximately solve this equation. In order to do so, we seek to minimize the

226

cost function below iteratively 227

J(xn+1

j)=(xn+1

j−xn

j)T(Cn

xx )−1(xn+1

j−xn

j)+g(xn+1

j)−d−qan+1en

jT(an+1Cdd )−1g(xn+1

j)−d−qan+1en

j,(11)

where 228

Nmda

∑

i=1

1

an=1.

We proceed as follows: we initially sample parameters

xj,0 ∼N(xf

,

Cxx )

and generate

229

ensemble of predicted observations

yj,0 =g(xj,0)

. We then use this ensemble to construct

230

covariance matrices

˜

Cn

yy

and

˜

Cn

xy

. We continue by perturbing observations, one for each

231

member 232

dn

j=d+en

j,en

j∼N(0, an+1Cdd). (12)

We now update each member according to 233

xn+1

j=xn

j+˜

Cn

xy ˜

Cn

yy +an+1Cdd −1(dn

j−yn

j). (13)

We ﬁnally forcast with updated parameters using 234

yn+1

j=g(xn+1

j),

and repeat to

Nmda −

1 steps. For simplicity, we summarize the ensemble methods used

235

below 236

•

We start by sampling a large ensemble of realizations of the prior uncertain parameters,

237

given their prescribed ﬁrst-guess values and standard deviations. 238

•

We then integrate the ensemble of model realizations forward in time to produce a

239

prior ensemble prediction, which also characterizes the uncertainty. 240

•

We compute the posterior ensemble of parameters by making use of the misﬁt between

241

prediction and observations, and the correlations between the input parameters and

242

the predicted measurements. 243

•

Ultimately, we compute the posterior ensemble prediction by a forward ensemble

244

integration. The posterior ensemble is then the “optimal" model prediction with the

245

ensemble spread representing the uncertainty. 246

More about ESMDA can be found in [

26

] and recent work using parameter estimation and

247

ESMDA can be found in [

27

]. The code we use to preform the ESMDA analysis is modiﬁed

248

from code developed by Dr. Geir Evensen to preform and ESMDA analysis with a SEIR

249

epedimic model. [28]250

2.3.3. Twin Model Analysis 251

This section describes the process through which the data assimilation models were

252

tuned, assessed, and used to provide insights into key stratospheric circulation drivers.

253

We determine appropriate values through twin model experiments in which data from

254

a known “truth” is assimilated. This is an important ﬁrst test of the data assimilation

255

system that has been widely implemented over several decades and disciplines [

29

–

32

].

256

For the particle ﬁlter, twin model experiments were used to determine appropriate ranges

257

for hyperparameters including assimilation period (number of days between observations

258

and associated updates), observation error, and ensemble size. In the context of ESMDA

259

identical twin experiments were used to assess the inﬂuence of decorrelation lengths. 260

First, identical twin model experiments were employed, using synthetic data from

261

simulating the underlying model (Eqn (1)-(3) with ﬁxed

h=

68 m, and

Λ

as in (4) with

262

Λ0=

0.75 m/s/km and

Λa=

2.25 m/s/km, and

ϵ=

0.3. Mean-zero Gaussian noise

263

Version September 30, 2023 submitted to Meteorology 8

with variance

σ2

obs

was added to the variable

U

and then assimilated as the observed

264

data. The data assimilation schemes were then applied with varying combinations of

265

hyperparameters and statistics computed to compare the ensemble distribution to the

266

known truth. We took the mean of the ensemble members as the assimilation analysis and

267

compared to the truth via the root mean squared error (RMSE) 268

RMSE =v

u

u

t1

N

N

∑

t=1Uens(t)−Utruth (t)2. (14)

Note that for the particle ﬁlter analysis, RMSE is computed on the dimensionalized

269

values and only over the last 10 years of assimilation, thereby considering the ﬁrst 10 years

270

as a spin-up before the analysis. In a similar way we assessed the recovery of known

271

parameters h,Λ0, and Λaunder various hyperparameters using RMSE. 272

Fraternal twin model experiments were used to assess the DA scheme’s ability

273

to recover data that comes from a model different than that being implemented in the

274

algorithm, under various hyperparaemters. Here, the ERA-Interim reanalysis wind speed

275

data is smoothed and considered the “truth” against which to compare the analysis mean

276

of ensemble distribution. The RMSE is again calculated to measure goodness of ﬁt. 277

3. Results 278

3.1. Particle Filter 279

3.1.1. Identical Twin Model Experiments 280

As described in Section 2.3.3, identical twin experiments were run on other hyperpa-

281

rameters including observation error, ensemble size, and assimilation period. Comparing

282

RMSE for varying ensemble size,

nens

, shows a common pattern of decreasing until some

283

critical size, after which increases in ensemble size no longer improve the estimation. Thus,

284

it is sufﬁcient and efﬁcient to choose an ensemble size that is just larger than the value at

285

which RMSE ceases to decrease. The ensemble size is ﬁxed at

nens =

300 for subsequent

286

analyses. 287

Assimilation period refers to the length of time between consecutive observations

288

of the system/updates of the ensemble. RMSE behaves intuitively for ﬁxed assimilation

289

period, and variable observation error and is demonstrated in (vertical slices of) Fig. 2.

290

Indeed, when the observations are very accurate (small

σ2

obs

) the particle ﬁlter is more

291

successful estimating the truth (small RMSE). However, as the observation error increases,

292

so does the measure of error of the ensemble average. Note that the observation error

293

corresponds to the non-dimensionalized wind speed data, which has a maximum of 1.08. 294

The relationship between RMSE and assimilation period, however, is a bit more nu-

295

anced, particularly for parameter estimation (for a ﬁxed observation error, this corresponds

296

to horizontal slices in Fig. 2). For state variable

U

, corresponding to zonal wind speed, it

297

appears that the decreased number of observations yields to less accurate state estimations,

298

particularly for large observation errors. However, this is not universally true, and even

299

using just 2% of the available observations with an assimilation period of 50 days can still

300

lead to nearly as good wind speed estimations when σ2

obs is small. 301

On the other hand, accuracy of parameter recovery appears to increase with increasing

302

assimilation periods for

h

,

Λ0

and

Λa

. Indeed, Fig. 2shows that for a ﬁxed observation

303

error, RMSE of parameter estimates generally decreases with longer time between observa-

304

tions/updates. This pattern is most prominent for estimation of h.305

3.1.2. ECMWF Data Assimilation 306

It is now of interest to investigate the use of the particle ﬁlter to gain an understanding

307

of stratospheric dynamics when “real-world” observations are used. To this end, fraternal

308

twin experiments, are conducted where the “truth” is a smoothed version of the observed

309

ECMWF mean zonal wind data described in Section 2.2. Noise is added to be used as

310

observations and the success of the particle ﬁlter at uncovering the smoothed data can be

311

Version September 30, 2023 submitted to Meteorology 9

Figure 2. RMSEs for combinations of assimilation period and observation error when the state vector

and control parameters hand Λare estimated simultaneously.

assessed. Thus, these experiments will investigate the ability of the model as well as the

312

particle ﬁlter to produce an analysis at least qualitatively similar to what is observed. 313

Results for varying assimilation periods and observation errors are shown in Fig. 3.

314

We now not only show RMSE for state variable

U

, but also estimate parameters

h

,

Λ0

,

315

and

Λa

. We note similarities in the RMSE proﬁles in these experiments with the identical

316

twin experiments (Fig. 2). However, the effects of increased assimilation periods are no

317

longer as apparent as in the twin model experiments. Now, to achieve the lowest RMSE

318

for windspeeds, it is best to assimilate all the data, and make daily updates. However,

319

using our insights from the identical twin experiments, we expect that a reasonable value

320

of

h

should be chosen from results with larger assimilation periods. Thus, Fig. 3may

321

suggest an initial Rossby wave amplitude

h≈

100 m. This is signiﬁcantly larger than

322

the

h=

68 m ﬁxed in the Ruzmaikin model. Similarly, we expect

Λ0≈

0.2 m/s/km

323

and

Λa≈

1.2 m/s/km, compared to the values in Ruzmaikin of 0.75 m/s/km and 2.25

324

m/s/km, respectively. 325

3.1.3. Summary 326

An exploration of the particle ﬁlter was initially motivated by the expected bimodality

327

of the zonal winds ensemble due to the bistability of the underlying model. We found that

328

signiﬁcant bimodality of the ensemble is achieved only for longer assimilation periods, also

329

corresponding to improved parameter estimates in identical twin experiments. With short

330

assimilation periods, frequent updates of the state variable

U

do not allow for the ensemble

331

to spread out and sample both stable branches of the

(h

,

U)

bifurcation diagram. Further, it

332

results in inferior parameter estimation, as the effects of the parameters are suppressed,

333

with the analysis being driven by the observations of U.334

Applying these ideas to the ECMWF reanalysis data, we obtain estimates for

h

within

335

the region of bistability when we assume longer assimilation periods. For short assimilation

336

periods,

h

is estimated as unrealistically small, again from the assimilation analysis being

337

driven by the daily state updates, requiring tropospheric perturbations to play a less

338

Version September 30, 2023 submitted to Meteorology 10

Figure 3. Estimates for

h

and the coefﬁcients of

Λ

when they are estimated simultaneously. (Top,

Left) the RMSE for mean zonal wind using the particle ﬁlter with ECMWF reanalysis data.

important role. However, even with updated parameter estimates, important phenomena

339

including spikes in winter winds, are largely missed in our data assimilation. Thus, we ﬁnd

340

that a constant

h

is insufﬁcient to capture the complex dynamics of the ECMWF reanalysis

341

data, as its effects are being suppressed for short assimilation periods, and unable to match

342

the data for longer assimilation periods. 343

With this in mind, we turned to ESMDA, which avoids bias from updates to the state

344

variables while also allowing for more ﬂexibility in the parameter estimations, including a

345

time-dependent h(t)and unparameterized Λ(t).346

3.2. ESMDA Analysis 347

3.2.1. Free Parameters 348

Here we discuss the parameters we are able to estimate using the ESMDA scheme

349

outlined in Sec 2.3.2. We have several distinct scenarios to investigate with our parameter 350

estimation. First, when

Λ(t)

is parameterized as in Eqn. 4with

Λ0

,

Λa

,

ϵ

, and

h

constant

351

and unknown. Second, when

h

is constant and unknown but

Λ(t)

, is replaced with a

352

vector,

Λ(t)∈RN

, where

N

is the number of days over which we have reanalysis data.

353

Third, when

Λ(t)

is as in the ﬁrst case but

h

is replaced with a vector,

h(t)∈RN

, for a time

354

dependent perturbation parameter and fourth, when both

Λ(t)∈RN

and

h(t)∈RN

so

355

that they are both time dependent. 356

For each of the scenarios above we also estimate the initial conditions for

X

,

Y

, and

357

U

. In the cases where we allow for time-dependent

Λ

or

h

, we also have the choice of

358

a decorrelation time

τλ

and

τh

. These parameters control how the vectors

Λ(t)

and

h(t)359

are sampled. The initial sampling of these parameters is done by randomly sampling

360

amplitudes and phases of sine and cosine terms penalizing shorter wavelengths according

361

to a negative exponential and the decorrelation time described in [

33

,

34

]. The longer the

362

decorrelation time, the more the shorter wavelengths are penalized in the sampling. As a

363

result, the longer the decorrelation time the smoother the time continuous priors for

Λ(t)364

Version September 30, 2023 submitted to Meteorology 11

and

h(t)

will be. The decorrelation lengths are not estimated by the ESMDA analysis but

365

speciﬁed beforehand. 366

For the ﬁrst case outlined above we introduce an additional two shift parameters that

367

will be estimated by the ESMDA analysis

cΛa

and

cϵ

which shift the sine functions in Eqn.

368

4to align with the data. That is Λ(t)becomes, 369

Λ(t) = Λ0+Λasin2π(t−cΛa)

1 year +ϵΛ0sin2π(t−cϵ)

11 year . (15)

We also have the freedom to assign observation errors to the reanalysis data. For the

370

experiments reported on below, we set the standard deviation of the error in the winds as 371

σu=

10

m/s

. We found in the identical twin model experiments, described below, that a

372

larger observation error provided better estimates of

Λ(t)

than with very low observation

373

errors. A relatively large observation error also allows for parameter estimates to not be

374

overly biased and avoid spurious overﬁtting. We also found improvements in RMSE stop

375

after about 1000 ensemble members which is what we use for all ESMDA experiments in

376

conjunction with 32 ESMDA steps. 377

3.2.2. Identical Twin Model Experiments 378

To evaluate the ESMDA scheme we ﬁrst run a series of twin model experiments

379

where our observational data is produced directly from our low order model. For this

380

example we generate observational values of

U

using a ﬁxed perturbation of

h=

68 m

381

and a

Λ(t)

parameterized as in Eqn. (4) with

ϵ=

0.3 and

Λ0

,

Λa

as prescribed in [

10

].

382

For these experiments we assume no prior structure on

Λ(t)

or

h(t)

and run ESMDA

383

experiments across the chosen combinations of

τλ={

1.5, 1, 1

/

2, 1

/

4, 1

/

12, 1

/

24

}

yr and

384

τh={

1.5, 1, 1

/

2, 1

/

4, 1

/

12, 1

/

24

}

yr. We use these same values for each of the other

385

experiments when applicable. For simplicity, we denote these values in the manuscript

386

with the closest integer number of days that they represent, speciﬁcally 547, 365, 182, 91, 30,

387

and 15 days. 388

The results are summarized in Fig. 4where we show the root mean squared error

389

(RMSE) for both

U(t)

and

Λ(t)

as well as the time-averaged value of

h(t)

for which a good

390

analysis should return a value close to

h=

68 m. The summary results show that the best

391

values for

τλ

are 547 and 365 days, while results do not depend as much on

τh

, evidenced by

392

the very similar values of RMSE for

U(t)

and

Λ(t)

for ﬁxed values of

τλ

. It is also important

393

to note that the lowest values of RMSE for

U(t)

correspond to the lowest values of RMSE

394

for

Λ(t)

and the best average values of

h(t)

despite only being conditioned on

U(t)

. This

395

establishes, at least for these twin model experiments, some level of uniqueness in the

396

analysis solution and a low risk of spurious parameter estimations which still produce

397

good ﬁts. For these experiments, we would expect the longer decorrelation lengths for

Λ(t)398

would produce better results as the period of the

Λ(t)

which produced the data is one year.

399

As will be discussed later, this is not the case when assimilating the reanalysis data which

400

may suggest more variation in the real Earth system. 401

In Fig. 5we show three examples of the analyses we obtain for both

Λ(t)

and

U(t)

as

402

well as the

(Λ

,

U)

phase space with the equilibrium solutions of the autonomous version

403

of the system in [

10

]. The best ﬁts for both

Λ(t)

and

U(t)

occur when

τλ=

547 which

404

produces a phase space very similar to that in [

10

] for the same parameters used to generate

405

the truth run. As

τλ

decreases, increased variation in

Λ(t)

is observable which translate

406

into the phase space. The higher values of

τλ

also provide time-averaged values for

h(t)407

closest to the true value of h=68 m. 408

While we only show the case where both

Λ(t)

and

h(t)

are free and time-dependent,

409

other runs where

h

was kept constant but estimated with the ESMDA scheme show similar

410

behavior typically obtaining analyses with

h≈

68 m for longer

τλ

. Truth run cases where

h411

was set to 68 m and only

Λ(t)

was estimated were also carried out, in this case, there was

412

no signiﬁcant improvement in analyses over allowing both parameters to be free. 413

Version September 30, 2023 submitted to Meteorology 12

(a) RMSE U(b) RMSE Λ(t)(c) Mean h(t)

Figure 4. Summary Statistics of Twin Model Experiment

These twin model experiments establish three important points. First, the ESMDA

414

scheme can successfully estimate parameters despite the high dimensionality resulting

415

from time-dependent

Λ(t)

and

h(t)

. Second, the best results as measured by the RMSE

416

between analysis mean zonal wind and the true wind corresponds to the most accurate

417

estimations of

Λ(t)

and

h(t)

rather than spurious parameter values that happen to produce

418

good mean zonal wind analyses. Third, the decorrelation lengths have a strong effect on

419

the quality of the analysis and must be considered. To that end, we range over the same

420

values for the decorrelation lengths for all experiments using the ECMWF reanalysis data 421

when a parameter is chosen to be time-dependent. 422

3.2.3. Parameterized Λ(t)and Free h(t)423

In this section we explore whether or not only a time-dependent

h(t)

can account

424

for the variability observed in the ECMWF reanalysis data. In the reduced order models

425

described in [

10

], [

8

], and [

9

], recall that the perturbation parameter

h

is used to represent

426

the effects of Rossby waves on the polar vortex. In these studies, a low value for

h

implies

427

weaker Rossby waves and thus higher mean zonal winds while a large value of

h

gives

428

rise to larger perturbations and lower mean zonal winds. In the autonomous version of the

429

system in analized in [

10

] there is a region of bistability between values of

h≈

25 m and

430

h≈

150 m. We may expect to see

(h

,

U)

values close to the equilibrium branches shown

431

in [

10

, Fig. 3] in phase space after assimilation of the ECMWF data, however this is not

432

what we typically see. The reason for this is that the mean zonal winds are typically much

433

smaller in the data than those coming from the chosen parameters in the reduced order

434

models. 435

Indeed, the bifurcation diagram for

h

in [

10

, Fig. 3] results from ﬁxing

Λ=

1 m/s/km.

436

However, as ﬁxed

Λ

decreases, the region of bistability shrinks, until the stable branches

437

converge to a single stable equilibrium around approximately

Λ≈

0.5 m/s/km, also

438

reﬂected in [

10

, Fig. 1]. In our ﬁt,

Λ

varies seasonally and is close to zero or negative during

439

the summer. 440

Here, we take

Λ(t)

as in Eqn. 15 but allow

Λ0

,

Λa

, and

ϵ

to be free parameters estimated

441

by ESMDA as well as the shift parameters,

cΛa

and

cϵ

. Analysis results are shown in Fig.

442

6. We ﬁnd that the amplitude of

Λ(t)

is much lower than the idealized cases explored in

443

the various reduced order models. This can be observed in Fig. 6e where we see the phase

444

space concentrated around

Λ=

0. In all the assimilations for this case, recovery of the

445

peaks in mean zonal wind was not achievable even when

h(t)

had a very low decorrelation

446

length to allow for more variability, which for this case corresponds to the lowest RMSE,

447

shown in Fig. 6a where we also note the the variance of

U

decreases with increasing

τλ

as

448

one might expect. In Fig. 6b we see that the variance in

h(t)

is correlated to the mean value

449

with typically increasing variance as

τh

increases. In this case, when

h(t)

is allowed to

450

vary quickly in time, smaller changes are needed to cause variation in

U

while an

h(t)

that

451

cannot vary rapidly in time tends to need larger amplitude changes to affect

U

. The analysis

452

curve can be compared to the observation in Fig 6d where we see good agreement in the

453

troughs but a missing of the peaks. The resulting

Λ(t)

from the ﬁt parameters, estimated

454

Version September 30, 2023 submitted to Meteorology 13

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 5. Examples from the twin model experiments. Left column:

τλ=

547,

τh=

91. Middle

column:

τλ=

182,

τh=

91, Right column:

τλ=

15,

τh=

91. Note: (g)-(h) only showcases the stable

equilibrium branches. This is also repeated in the subsequent Figures.

(a) (b)

(c) (d) (e)

Figure 6. Lowest RMSE Experiment (h=15) for Case 1 (Λ0,Λa, and ϵfree)

Version September 30, 2023 submitted to Meteorology 14

(a) (b)

Figure 7. Summary Statistics of Free Λ(t)and constant h(t)

h(t)

and

U

are shown in Fig. 6c. The example of assimilation results corresponds to

τh=

15

455

chosen because it provides the lowest RMSE value with mean zonal winds. 456

Due to the relatively small estimated amplitude of

Λ

and the inability to capture

457

wintertime windspeeds, we also explored enforcing larger values for

Λa

. However, these

458

assimilation experiments produced higher RMSE values by a factor of two, overshot the

459

summertime troughs, and still failed to “jump” to the upper branch of the bifurcation

460

diagram. The full results of this endeavor are presented in Appendix C.461

3.2.4. Free Λ(t)and Constant h462

In the previous section we observed that a time-dependent perturbation parameter

463

h(t)

alone was insufﬁcient to allow the Ruzmaikin model to accurately capture the ECMWF

464

renalysis data. In this section, we describe the results of our ESMDA experiments where

465

we take

h

to be constant and estimate an unparameterized

Λ(t)

. We note that taking a

466

constant

h

still allows for perturbations to the system and the estimation of a constant

h467

is still informative in examining how well a reduced order model can represent realistic

468

data. In general, we ﬁnd that with only a time-dependent

Λ(t)

estimated by the ESMDA

469

scheme we achieve very good ﬁts to the reanalysis data with the best ﬁts occurring for

470

lower values of

τλ

. This is in contrast to our truth runs where longer decorrelation lengths

471

produced the best ﬁts. However for those runs the prescribed

Λ(t)

was slowly changing

472

and correlated over long times being represented by a

sin

wave with a period of 1 yr. The

473

fact that the lowest RMSE values occur for smaller

τλ

suggest more variability in

Λ(t)

is

474

needed to account for some of the rapid changes in the renalysis data. We show summary

475

results for these experiments in Fig. 7where we see a general trend of increasing RMSE in

476

U(t)with decreasing variance in Λ(t).477

In Fig. 8we show the results of the ESMDA parameter estimation for

τλ= [

547, 182, 15

]478

days. All show relatively good agreement with the reanalysis data with the smaller values

479

of

τλ

providing the best ﬁt to the extremes of mean zonal winds in the data Figs. 8f,8e,

480

and 8d. In Figs. 8c,8b, and 8a corresponding to

τλ=

30, 182 and 547 respectively, we see a

481

general trend of diminishing peaks and deeper troughs in

Λ(t)

as

τλ

decreases. This trend

482

also emerges in Fig. 7b as the mean of

Λ(t)

sharply decreases after

τλ=

91. Interestingly

483

it is for these cases that the peaks of the mean zonal wind are best captured despite the

484

generally lower values of

Λ(t)

. This may have to do with the ability of a more rapid and

485

dynamic recovery in

Λ(t)

after a large dip permitted when

τλ

is small. We also ﬁnd that

486

reletavily low values of

h

are estimated with

h≈

35 m in these cases. This is likely due to

487

the generally lower mean zonal winds represented in the data as opposed to the idealized

488

cases examined in [10]. 489

In Figs. 8i,8h, and 8g we show the

(Λ(t)

,

U(t))

phase space with the equilibrium

490

solutions of the autonomous system in [

10

]. The phase space orbits the equilibrium solu-

491

tions with the most time spent on the lower branch and some jumps to the upper stable

492

Version September 30, 2023 submitted to Meteorology 15

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 8. Examples with

Λ(t)

free and

h

constant. Left column:

τλ=

547. Middle column:

τλ=

182,

Right column: τλ=30.

branch. The fact that the phase space is so similar to the idealized time dependent case

493

in [

10

] demonstrates that the parameters required to accurately match the reanalysis data

494

are not far ﬁeld from the idealized case. This also suggests that the reduce order model

495

captures the most important physics of the real system. 496

3.2.5. Free Λ(t)and h(t)497

Here, we allow both parameters to be free as in the identical twin model experiments.

498

Like in the twin model case, we typically ﬁnd that RMSE for the mean zonal wind is not

499

very dependent on

τh

but strongly dependent on

τλ

as can be seen in Fig. 9a. Unlike the

500

twin model case, the lowest values for mean zonal RMSE occur for lower values of

τλ

, this

501

suggests that more variability in

Λ(t)

is required to match the ECMWF reanalysis data. As

502

might be expected as

τλ

decreases the variance in both

Λ(t)

and

U(t)

increase, shown in

503

Figs. 9c and 9b. In Fig. 9d we see the variance in

h(t)

decreases as

τλ

decreases suggesting

504

that the model is most sensitive to

Λ(t)

. It is for the cases that

h(t)

varies less and

Λ(t)505

varies more we achieve the lowest values in RMSE for the mean zonal wind. In all cases we

506

are able to capture the peaks fairly well and extremely well for small τλ.507

In Fig. 10 we show three examples from these experiments for a ﬁxed

τh

as the results

508

were not sensitive to that parameter. It is is also notable how similar these results are to

509

those in the previous section where

h

was ﬁxed, a visual inspection between Fig. 8and 10

510

shows this with nearly identical results and extremely similar phase spaces. 511

In Figs. 10a,10b, and 10c we see that the estimated

h(t)

takes on a wider range of

512

values when

τλ

is large than it does when

τλ

is small. Allowing

h

to be time-dependent

513

does provide slightly smaller RMSE values but not by much, and as

Λ(t)

is allowed to be

514

more dynamic it appears h(t)needs to do "less work" for a good data match. 515

Version September 30, 2023 submitted to Meteorology 16

(a) RMSE U(b) UVariance

(c) Λ(t)Variance (d) h(t)Variance

Figure 9. Summary Statistics of Free Λ(t)and h(t)

3.3. Analysis Around SSW Events 516

In furtherance of our analysis on how representative reduced order models of the

517

polar vortex can be, we compare our analysis curves to a list of known sudden stratospheric

518

warming (SSW) events taken from [

35

]. To do this we take snapshots of the analysis curves

519

28 days before and after the speciﬁed SSW event looking at

Λ(t)

and

U(t)

. We would

520

expect to see a sudden drop in

Λ(t)

corresponding to a rapid reduction in the thermal

521

gradient between the lower and upper latitudes in the stratosphere followed by a drop in

522

mean-zonal winds with a slow or very little recovery in the 28 days after the event. This

523

behavior is inline with timelines for and deﬁnitions of various events dubbed SSW events

524

[36]. 525

In general, this pattern is what we observe throughout all of our experiments where

526

Λ(t)

is estimated completely by the ESMDA scheme. The pattern described is most

527

noticeable for longer

τλ

but does persist for shorter decorrelation lengths while not being

528

sensitive to

τh

. In Fig. 11, we show examples of these snapshots for three selected dates

529

with

τλ=

547, 182, and 15 and

τh=

365. For the majority of the SSW events examined,

Λ(t)530

exhibits a convex shape with a minimum near the listed date of the SSW and a recovery

531

thereafter. The mean zonal wind exhibits a delayed decrease with no real recovery in the

532

28 day period after the listed SSW date. When

τλ

is small enough the

Λ(t)

becomes less

533

smooth and more dynamic but still exhibits the same general pattern. We also see that for

534

smaller

τλ

the minimum value of

Λ(t)

becomes more extreme for the majority of cases.

535

This is because for small decorrelation lengths

Λ(t)

is not tied to values far in the past or

536

future and can respond quickly to rapid changes in the data. We show the snapshots for all

537

dates listed in [

35

] in Figures A3 (for

τλ=

547), A4 (for

τλ=

182), and A5 (for

τλ=

15).

538

There it is noteworthy that the general pattern observed in these snapshots only emerges

539

for shorter τλon the March, 24, 2010, SSW event. 540

These results are interesting as they show the reduced order model parameters respond

541

appropriately to real physical events. 542

Version September 30, 2023 submitted to Meteorology 17

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 10. Examples with both

Λ(t)

and

h(t)

free. Left column:

τλ=

547,

τh=

365. Middle column:

τλ=182, τh=365, Right column: τλ=15, τh=365

Version September 30, 2023 submitted to Meteorology 18

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 11. Examples of SSW snap shots for: top row

τλ=

547, middle row

τλ=

182, and bottom row

τλ=15

(a) 1 year window (b) 2 year window (c) 5 year window

(d) 1 year window (e) 2 year window (f) 5 year window

Figure 12. Variance of Λ(t)over several different moving windows, 1 year, 2 years, and 5 years

Version September 30, 2023 submitted to Meteorology 19

Finally, we comment on some trends observed in the analysis data. We computed 1

543

year, 2 year, and 5 year moving averages and variances for the analysis

Λ(t)

and

U(t)

. In

544

general, there was a noticeable upward trend for both, which becomes more pronounced

545

for the longer windows. While these trends are not deﬁnitive, they may suggest shifts in

546

the behavior of the polar vortex over the 20-year data set considered. We show an example

547

of these trends in Fig. 12.548

4. Conclusions and Discussion 549

In this manuscript, we demonstrated that reduced order models of stratospheric

550

wave-zonal interactions, in particular the Ruzmaikin model (2003) in conjunction with data

551

assimilation schemes can be used to produce model output closely matches with averaged

552

ERA-Interim reanalysis data. We employed a 20-year dataset of atmospheric reanalysis data

553

sourced from the ECMWF for the purpose of understanding the behavior and inﬂuence of

554

physically-based control parameters of the low order model in the context of real-world

555

phenomena. Initially, we applied a particle ﬁlter due to the assumed bimodality of the

556

ensemble associated with bistability of the model and of the winter polar vortex. Yet, this

557

method ultimately failed to capture several relevant features of the data, and subsequently,

558

we utilized ESMDA techniques. 559

When using the particle ﬁlter, we determined the inﬂuence of hyperparameters

560

through identical twin model experiments. We conﬁrmed that lower observation error

561

results in better estimation of state variables and parameters. However, the effects of longer

562

assimilation periods was initially surprising, as we found that assimilating less data can

563

actually improve parameter estimation. One reason this may be the case is that the ODE

564

model is given more time to “feel” the inﬂuence of the parameter updates. With short

565

assimilation periods, the parameters show much larger variances, which may come from

566

over-ﬁtting the data. 567

We used fraternal twin experiments to determine how well the reduced order model

568

could recover ECMWF reanalysis data using the particle ﬁlter algorithm. We noted simi-

569

larities in the RMSE proﬁles with the identical twin experiments. However, even with the 570

updated parameter estimates, important phenomena related to the winter polar vortex

571

are largely missed in our data assimilation. Thus, ESMDA was employed to explore the

572

applicability of the Ruzmaikin model when

h

and

Λ

are allowed to be fully free to vary

573

with respect to time, rather than being prescribed in a ﬁxed form. 574

We had several distinct scenarios to investigate using ESMDA. First, when

Λ(t)

is

575

parameterized as in Eqn. 4with

Λ0

,

Λa

,

ϵ

, and

h

constant and unknown. Second, when

h

is

576

constant and unknown but

Λ(t)

is unparameterized and free. Third, when

Λ(t)

is as in the

577

ﬁrst case but

h

is replaced with an unparameterized, free vector,

h(t)

for a time-dependent

578

perturbation parameter and fourth, when both

Λ(t)

and

h(t)

are both time-dependent and

579

unparameterized. In the cases where we allowed for time-dependent

Λ

or

h

, we also had

580

the choice of a decorrelation time τλand τh.581

To evaluate the ESMDA scheme, we did ﬁrst run a series of twin model experiments, as

582

in the case of the particle ﬁlter, where our observational data is produced directly from our

583

low order model. Overall, using ESMDA, we concluded that a free

Λ(t)

and

h(t)

and sam-

584

pling the space of reasonable decorrelation lengths provided an improvement in data ﬁtting

585

with the best results coming from relatively long decorrelation lengths for

h

. Furthermore,

586

the recovered parameters which produce close matches to the reanalysis data are in line

587

with the idealized situations considered in [

10

] with the time-dependent

(Λ(t)

,

U(t))

phase

588

space orbiting the stable equilbrium branches of an even further simpliﬁed autonomous

589

version of the reduced order model. 590

We also examined our analysis curves around some known SSW events, ﬁnding a

591

general pattern in

Λ(t)

and

U(t)

consistent with what one would expect for such an event.

592

We also noticed a general increasing trend in the moving averages and variances of

Λ(t)593

and

U(t)

which may be a result of increasing tropospheric and decreasing stratospheric

594

temperatures resulting from increased CO

2

levels in the atmosphere. It has been under-

595

Version September 30, 2023 submitted to Meteorology 20

stood that increases in global CO

2

should lead to increases in tropospheric temperatures

596

and decreases in stratospheric temperatures as warmer surface pushes the edge of the

597

stratosphere higher [

37

]. Evidence of such a signature of increasing CO

2

levels has been

598

observed in data [

38

]. A cooler stratosphere may cause stronger temperature gradients

599

leading to generally higher mean zonal winds and more variability. 600

Author Contributions: 601

Project conceptualization and supervision by CKRTJ. ECMWF data processing and Fig 1by ZW.

602

Particle ﬁlter analysis and visualizations by JS. ESMDA core algorithm coded by ZW, with edits and

603

analysis by CS and EF, and visualizations by CS. Main manuscript written by JS, CS, and EF. Edits

604

and revisions by ZW and CKRTJ. All authors have read and agreed to the published version of the

605

manuscript. 606

Funding: EF was supported by NSF grant DMS-2137947 during the work on this research. CS, EF,

607

and CKRTJ were supported by ONR grant N000141812204. This project was inspired by work from

608

the 2019 MCRN Summer School, funded by NSF grant DMS-1722578. 609

Data Availability Statement: ECMWF ERA-Interim data utilized in this work was publicly available

610

at https://www.ecmwf.int/en/forecasts/dataset/ecmwf-reanalysis-interim, however, as of June

611

1, 2023, users are strongly advised to migrate to ERA5 from the Climate Data Store (CDS). Data

612

processing and analysis scripts are available from the corresponding author by reasonable request. 613

Acknowledgments: We would like to thank Dr. Geir Evensen for his advice in modifying his

614

original codes to ﬁt this particular problem. We would like to thank Dr. Mohamed Moustaoui for his

615

suggestions regarding appropriate and interesting atmospheric science interpretations and directions.

616

Finally, we would like to thank those who started this project with us at the 2019 MCRN Summer

617

School - Kiara Sanchez, Ligia Flores, and Yorkinoy Shermatova. 618

Conﬂicts of Interest: The authors declare no conﬂict of interest. 619

Appendix A 620

Table A1. Coefﬁcients of the Ruzmaikin model (1)-(3)

Parameter Value

τ1122.6276

r0.6286

s1.9638

ξ1.7488

δw70.8437

ζ240.5361

UR0.4748

τ230.3713

η9.131 ×104

δΛ4.9115 ×10−4

Λ00.75 m/s/km

δΛa2.25 m/s/km

ϵ0 - 0.3

Appendix B 621

This section reviews the mathematical formulation underlying the particle ﬁlter. Nota-

622

tion and outline borrowed from [

16

]. Then, the functions and variables are deﬁned in the

623

context of the model and data described in sections 2.1 and 2.2, respectively. 624

Let xk∈Rnbe the state vector which evolves according to the system model 625

xk+1=fk(xk,wk)(A1)

Version September 30, 2023 submitted to Meteorology 21

where

fk:Rn×Rm→Rn

and

wk∈Rm

is a zero-mean, white-noise sequence independent

626

of

x

. Let

yk∈Rp

represent observations which are related to

xk

through the observation

627

equation 628

yk=hk(xk,vk)(A2)

where

hk:Rp×Rr→Rp

is the observation operator and

vk∈Rr

is another zero mean,

629

white-noise sequence with known distribution independent of xkand wk.630

The goal of the particle ﬁlter is to construct the density of

xk

given all preceding

631

observations

Dk={yi:i=

1,

. . .

,

k}

. This is done via recursion and Bayes’ rule, which

632

states 633

p(xk|Dk) = p(yk|xk)p(xk|Dk−1)

p(yk|Dk−1). (A3)

Note that each term in this equation can be written as a function of known variables. Indeed,

634

the denominator is 635

p(yk|Dk−1) = Zp(yk|xk)p(xk|Dk−1)dxk(A4)

where the ﬁrst term in the integrand (and numerator of (5)) can be written as 636

p(yk|xk) = Zδ(yk−hk(xk,vk))p(vk)dvk. (A5)

The remaining term in the numerator of (5) can be decomposed similarly, 637

p(xk|Dk−1) = Zp(xk|xk−1)p(xk−1|Dk−1)dxk−1(A6)

again writing the ﬁrst term in the integrand as 638

p(xk|xk−1) = Zδ(xk−fk−1(xk−1,wk−1))p(wk−1)dwk−1. (A7)

Thus, Bayes’ rule can be rewritten in terms of known quantities and the recursively

639

deﬁned

p(xk−1|Dk−1)

. Analytical solutions to this problem are available for the constrained

640

case for linear

fk

,

hk

and Gaussian distributions. For more general applications, including

641

the one considered here, the following numerical algorithm is utilized instead. 642

Consider a set of random samples

{xk−1(i):i=

1,

. . . N}

of known distribution

643

p(xk−1|Dk−1). The prediction step involves calculating 644

x∗

k(i) = fk−1(xk−1(i),wk−1(i)) (A8)

where

wk−1(i)

is sampled from the known distribution

p(wk−1)

. Clearly,

{x∗

k(i)}

is dis-

645

tributed as p(xk|Dk−1).646

An update is then preformed by resampling

N

times with replacement according to

647

the discrete distribution where the weight for the i−th ensemble member is given by 648

qi=p(yk|x∗

k(i))

∑N

j=1p(yk|x∗

k(j)) , (A9)

to get {xk(i):i=1, . . . , N}which is distributed as p(xk|Dk)according to [24]. 649

In the application to stratospheric zonal winds studied here, we have the following

650

particle ﬁlter functions and distributions. The state vector is 651

xk=X Y U h Λ0ΛaT, (A10)

which evolves according to the dynamical model, fk, given by numerically solving (using 652

ode4 [

39

] in MATLAB) the system of ODEs (1)-(3) and evolving

X

,

Y

, and

U

according to

653

Version September 30, 2023 submitted to Meteorology 22

(a) (b)

(c) (d) (e)

Figure A1. Lowest RMSE Experiment (

τh=

91) for Case 2 (

Λ0=

0.75,

Λa=

2.25 m/s/km, and

ϵ=0.3 ﬁxed)

the solution, and adding model error,

wk∼N(

0,

Σmodel)

. The observed variable is

yk=U

,

654

related to state vector by the observation equation 655

yk=xT

ke3+vk(A11)

where

e3=001000T

, and observation error is assumed to mean-zero Gaussian

656

noise, vk∼N(0, σ2

obs).657

The covariance matrix of model error,

Σmodel

, is diagonal with elements set to 0.1%

658

of the maximum values for each state variable

X

,

Y

, and

U

. This was determined by

659

running the Ruzmaikin model with perturbations of varying magnitude between timesteps.

660

Ultimately, we chose the largest variance that allows for some “switching” between high

661

and low stable equilibrium wind speeds without totally disrupting the model. Note that this

662

also agrees fairly well with the method described in [

32

], of using the variance-covariance

663

matrix to estimate “natural” variation in a free run to use as an estimate of model variance.

664

Appendix C 665

Here, we further explore the ESMDA scheme applied to a parameterized

Λ(t)

as in

666

Eqn. 15 and a free

h(t)∈RN

. In Section 3.2.3, hereafter referred to as “case 1”, recall that

667

we were unable to hit the winter peaks using estimated

Λ0

,

Λa

,

ϵ

, and shift parameters.

668

Thus, we try to constrain these parameters to enforce larger values of

Λ

, so that it may

669

enter the region of bistability. 670

In Fig. A1, we ﬁx the parameters in Eqn. 15 except for the shift parameters and set

671

them to the values in [

10

] (case 2). We do this to enforce a larger amplitude for

Λ(t)

and

672

examine if a time dependent

h

is enough in this case to produce an analysis consistent with

673

the ECMWF data and the peaks represented in it. We note larger values in RMSE than in

674

case 1 shown in Fig. A1a however the inverse relationship between the RMSE and variance

675

of

U

in case one is no longer present. In Fig. A1b, we can see that for the lowest RMSE

676

results the average value of

h(t)

is highest, and higher than those in case 1. This results

677

from the larger amplitude of

Λ(t)

being enforced,

h

will generally need to be higher to

678

Version September 30, 2023 submitted to Meteorology 23

(a) (b)

(c) (d) (e)

Figure A2. Lowest RMSE Experiment (h=547) for Case 3 (Λ0and ϵfree, Λa≥2.25 m/s/km)

drive the resulting higher mean zonal winds down. In Fig. A1d we see that we are still

679

unable to capture the peaks despite the larger amplitude of

Λ(t)

evidenced in Fig. A1e. In

680

addition the troughs are overshot likely resulting from the large values of

h

required to

681

bring down mean zonal wind amplitudes. In Fig. A1c we show normalized

Λ(t)

,

U(t)

,

682

and

h(t)

for the case where

τh=

91, again corresponding to the lowest RMSE for mean

683

zonal wind. 684

Finally, with results shown in Fig. A2 we again use the ESMDA scheme to estimate

Λ0

,

685

Λa

, and

ϵ

, as well as the shift parameters, but provide a minimum value for

Λa

to be that

686

set in [

10

]. This is done to sample larger amplitudes in an attempt to ﬁnd an

h(t)

which can

687

allow for a close match of the data while capturing the peaks. The results are similar to

688

the previous case with the lowest RMSE occurring for

τh=

547. The estimated amplitudes

689

are indeed larger which can be seen in Fig. A2e however we are still unable to capture the

690

peaks and the troughs are also overshot as in case 2 shown in Fig. A2a. We also see larger

691

average values for h(t), Fig. A2b, than case 1 or 2 owing to the larger amplitudes in Λ(t).692

Appendix D 693

Here, we show the full results of ESMDA analysis behavior around SSW events for

694

varying decorrelation lengths. 695

696

Version September 30, 2023 submitted to Meteorology 24

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n)

Figure A3. Λ(t)and U(t)28 days around known SSW Events with τλ=547, τh=365.

Version September 30, 2023 submitted to Meteorology 25

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n)

Figure A4. Λ(t)and U(t)28 days around known SSW Events with τλ=182, τh=365.

Version September 30, 2023 submitted to Meteorology 26

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n)

Figure A5. Λ(t)and U(t)28 days around known SSW Events with τλ=15, τh=365.

Version September 30, 2023 submitted to Meteorology 27

References 697

1. Muench, H.S. On the dynamics of the wintertime stratosphere circulation. Journal of Atmospheric Sciences 1965,22, 349–360. 698

2.

Geisler, J. A numerical model of the sudden stratospheric warming mechanism. Journal of Geophysical Research 1974,79, 4989–4999.

699

3.

Charney, J.G.; DeVore, J.G. Multiple ﬂow equilibria in the atmosphere and blocking. Journal of Atmospheric Sciences 1979,

700

36, 1205–1216. 701

4.

Ambaum, M.H.P.; Hoskins, B.J. The NAO Troposphere–Stratosphere Connection. Journal of Climate 2002,15, 1969 – 1978.

702

https://doi.org/https://doi.org/10.1175/1520-0442(2002)015<1969:TNTSC> 2.0.CO;2.703

5.

Finkel, J.; Abbot, D.S.; Weare, J. Path properties of atmospheric transitions: illustration with a low-order sudden stratospheric

704

warming model. Journal of the Atmospheric Sciences 2020,77, 2327–2347. 705

6. Holton, J.R.; Mass, C. Stratospheric vacillation cycles. Journal of Atmospheric Sciences 1976,33, 2218–2225. 706

7.

Wakata, Y.; Uryu, M. Stratospheric multiple equilibria and seasonal variations. Journal of the Meteorological Society of Japan. Ser. II

707

1987,65, 27–42. 708

8. Yoden, S. Bifurcation properties of a stratospheric vacillation model. Journal of the atmospheric sciences 1987,44, 1723–1733. 709

9.

Yoden, S. An illustrative model of seasonal and interannual variations of the stratospheric circulation. Journal of the atmospheric

710

sciences 1990,47, 1845–1853. 711

10.

Ruzmaikin, A.; Lawrence, J.; Cadavid, C. A simple model of stratospheric dynamics including solar variability. Journal of climate

712

2003,16, 1593–1600. 713

11.

McIntyre, M.E. How well do we understand the dynamics of stratospheric warmings? Journal of the Meteorological Society of Japan.

714

Ser. II 1982,60, 37–65. 715

12.

Scott, R.; Polvani, L.M. Internal variability of the winter stratosphere. Part I: Time-independent forcing. Journal of the Atmospheric

716

Sciences 2006,63, 2758–2776. 717

13.

Stan, C.; Straus, D.M. Stratospheric predictability and sudden stratospheric warming events. Journal of Geophysical Research:

718

Atmospheres 2009,114.719

14.

Jeppesen, J. Fact sheet: Reanalysis. URL https://www.ecmwf.int/en/about/media-centre/focus/2020/fact-sheet-reanalysis,

720

2020. Last accessed 2021-07-31. 721

15.

Kantas, N.; Doucet, A.; Singh, S.S.; Maciejowski, J.; Chopin, N. On particle methods for parameter estimation in state-space

722

models. Statistical science 2015,30, 328–351. 723

16.

Gordon, N.J.; Salmond, D.J.; Smith, A.F. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. In Proceedings of

724

the IEE Proceedings F-radar and signal processing. IET, 1993, Vol. 140, pp. 107–113. 725

17.

Asch, M.; Bocquet, M.; Nodet, M. Data assimilation: methods, algorithms, and applications; SIAM, Society for Industrial and Applied

726

Mathematics, 2016. 727

18.

Bocquet, M.; Sakov, P. An iterative ensemble Kalman smoother. Quarterly Journal of the Royal Meteorological Society 2013,

728

140, 1521–1535. https://doi.org/10.1002/qj.2236.729

19.

Carrassi, A.; Bocquet, M.; Bertino, L.; Evensen, G. Data Assimilation in the Geosciences - An overview on methods, issues and

730

perspectives, 2018, [arXiv:physics.ao-ph/1709.02798].731

20.

Skjervheim, J.; Evensen, G. An ensemble smoother for assisted history matching. SPE 141929, 2011. https://doi.org/10.2118/14

732

1929-ms.733

21.

Sanchez, K. Understanding the Stratospheric Polar Vortex: A Parameter Sensitivity Analysis on a Simple Model of Stratospheric

734

Dynamics. PhD thesis, Sam Houston State University, 2020. 735

22.

Berrisford, P.; Kållberg, P.; Kobayashi, S.; Dee, D.; Uppala, S.; Simmons, A.; Poli, P.; Sato, H. Atmospheric conservation properties

736

in ERA-Interim. Quarterly Journal of the Royal Meteorological Society 2011,137, 1381–1399. 737

23.

Wu, Z. Data Assimilation and Uncertainty Quantiﬁcation with Reduced-order Models. PhD thesis, Arizona State University,

738

2021. 739

24.

Smith, A.F.; Gelfand, A.E. Bayesian statistics without tears: a sampling–resampling perspective. The American Statistician 1992,

740

46, 84–88. 741

25.

Marín, D.A.A. Particle ﬁlter tutorial. https://www.mathworks.com/matlabcentral/ﬁleexchange/35468-particle-ﬁlter-tutorial,

742

2012. 743

26.

Emerick, A.A.; Reynolds, A.C. Ensemble smoother with multiple data assimilation. Computers & Geosciences 2013,55, 3–15.

744

Ensemble Kalman ﬁlter for data assimilation, https://doi.org/https://doi.org/10.1016/j.cageo.2012.03.011.745

27.

Fleurantin, E.; Sampson, C.; Maes, D.P.; Bennett, J.; Fernandes-Nunez, T.; Marx, S.; Evensen, G. A study of disproportionately

746

affected populations by race/ethnicity during the SARS-CoV-2 pandemic using multi-population SEIR modeling and ensemble

747

data assimilation. Foundations of Data Science 2021,3, 479–541. https://doi.org/10.3934/fods.2021022.748

28. Evensen, G. geirev/EnKF_seir; 2021. 749

29.

Lawson, L.M.; Spitz, Y.H.; Hofmann, E.E.; Long, R.B. A data assimilation technique applied to a predator-prey model. Bulletin of

750

Mathematical Biology 1995,57, 593–617. 751

30.

Houtekamer, P.L.; Mitchell, H.L. Data assimilation using an ensemble Kalman ﬁlter technique. Monthly Weather Review 1998,

752

126, 796–811. 753

31.

Freeman, J.B.; Dale, R. Assessing bimodality to detect the presence of a dual cognitive process. Behavior research methods 2013,

754

45, 83–97. 755

Version September 30, 2023 submitted to Meteorology 28

32.

Browne, P.; Van Leeuwen, P. Twin experiments with the equivalent weights particle ﬁlter and HadCM3. Quarterly Journal of the

756

Royal Meteorological Society 2015,141, 3399–3414. 757

33.

Evensen, G. The Ensemble Kalman Filter: theoretical formulation and practical implementation. Ocean Dynamics 2003,53, 343–367.

758

https://doi.org/10.1007/s10236-003-0036-9.759

34.

Evensen, G. Sampling strategies and square root analysis schemes for the EnKF. Ocean Dynamics 2004,54, 539–560. https:

760

//doi.org/10.1007/s10236-004-0099-2.761

35.

Butler, A.H. Table of major mid-winter SSWs in reanalyses products. NOAA Chemical Science Laboratory, 2020. https://csl.noaa.

762

gov/groups/csl8/sswcompendium/majorevents.html.763

36.

Butler, A.H.; Seidel, D.J.; Hardiman, S.C.; Butchart, N.; Birner, T.; Match, A. Deﬁning Sudden Stratospheric Warmings. Bulletin of

764

the American Meteorological Society 2015,96, 1913–1928. https://doi.org/10.1175/bams-d-13-00173.1.765

37.

Manabe, S.; Wetherald, R.T. Thermal Equilibrium of the Atmosphere with a Given Distribution of Relative Humidity. Journal of

766

the Atmospheric Sciences 1967,24, 241–259. https://doi.org/10.1175/1520-0469(1967)024<0241:teotaw> 2.0.co;2.767

38.

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

768

stratospheric contribution to human ﬁngerprints on atmospheric temperature. Proceedings of the National Academy of Sciences 2023,

769

120.https://doi.org/10.1073/pnas.2300758120.770

39.

Keevil, J. ODE4 gives more accurate results than ODE45, ODE23, ODE23s. https://www.mathworks.com/matlabcentral/

771

ﬁleexchange/59044-ode4-gives-more-accurate-results-than-ode45-ode23-ode23s, 2016. 772

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual

773

author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to

774

people or property resulting from any ideas, methods, instructions or products referred to in the content. 775