Content uploaded by Emmanuel Fleurantin
Author content
All content in this area was uploaded by Emmanuel Fleurantin on Oct 03, 2023
Content may be subject to copyright.
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
