The widely accepted existence of an inherent limit of atmospheric predictability is usually attributed to weather’s sensitive dependence on initial conditions. This signature feature of chaos was first discovered in the Lorenz system, initially derived as a simplified model of thermal convection. In a recent study of a high-dimensional generalization of the Lorenz system, it was reported that the predictability of its chaotic solutions exhibits a non-monotonic dimensional dependence. Since raising the dimension of the Lorenz system is analogous to refining the model vertical resolution when viewed as a thermal convection model, it is questioned whether this non-monotonicity is also found in numerical weather prediction models. Predictability in the sense of sensitive dependence on initial conditions can be measured based on deviation time, that is, the time of threshold-exceeding deviations between the solutions with minute differences in initial conditions. Through ensemble experiments involving both the high-dimensional generalizations of the Lorenz system and real-case simulations by a numerical weather prediction model, this study demonstrates that predictability can depend non-monotonically on model vertical resolution. Further analysis shows that the spatial distribution of deviation time strongly contributes to this non-monotonicity. It is suggested that chaos, or sensitive dependence on initial conditions, leads to non-monotonic dependence on model vertical resolution of deviation time and, by extension, atmospheric predictability.

1.
S.
Park
, “
A unified convection scheme (UNICON). Part I: Formulation
,”
J. Atmos. Sci.
71
,
3902
3930
(
2014
).
2.
T.
Ahmed
,
H.-G.
Jin
, and
J.-J.
Baik
, “
A physically based raindrop–cloud droplet accretion parametrization for use in bulk microphysics schemes
,”
Q. J. R. Meteorol. Soc.
146
,
3368
3383
(
2020
).
3.
E.
Kalnay
,
Atmospheric Modeling, Data Assimilation and Predictability
(
Cambridge University Press
,
2003
).
4.
P.
Bauer
,
A.
Thorpe
, and
G.
Brunet
, “
The quiet revolution of numerical weather prediction
,”
Nature
525
,
47
55
(
2015
).
5.
S. G.
Benjamin
,
J. M.
Brown
,
G.
Brunet
,
P.
Lynch
,
K.
Saito
, and
T. W.
Schlatter
, “
100 years of progress in forecasting and NWP applications
,”
Meteorol. Monogr.
59
,
13.1
13.67
(
2019
).
6.
F.
Zhang
,
Y. Q.
Sun
,
L.
Magnusson
,
R.
Buizza
,
S.-J.
Lin
,
J.-H.
Chen
, and
K.
Emanuel
, “
What is the predictability limit of midlatitude weather?
,”
J. Atmos. Sci.
76
,
1077
1091
(
2019
).
7.
E. N.
Lorenz
, “
Deterministic nonperiodic flow
,”
J. Atmos. Sci.
20
,
130
141
(
1963
).
8.
E. N.
Lorenz
, “
The predictability of a flow which possesses many scales of motion
,”
Tellus
21
,
289
307
(
1969
).
9.
T. N.
Palmer
,
A.
Döring
, and
G.
Seregin
, “
The real butterfly effect
,”
Nonlinearity
27
,
R123
R141
(
2014
).
10.
E. N.
Lorenz
, “
Atmospheric predictability as revealed by naturally ocurring analogues
,”
J. Atmos. Sci.
26
,
636
646
(
1969
).
11.
S.
Moon
,
J. M.
Seo
, and
J.-J.
Baik
, “
High-dimensional generalizations of the Lorenz system and implications for predictability
,”
Phys. Scr.
95
,
085209
(
2020
).
12.
A. N.
Kolmogorov
, “
The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers
,”
Dokl. Akad. Nauk SSSR
30
,
299
303
(
1941
).
13.
R. H.
Kraichnan
, “
Inertial ranges in two-dimensional turbulence
,”
Phys. Fluids
10
,
1417
(
1967
).
14.
J. J.
Tribbia
and
D. P.
Baumhefner
, “
Scale interactions and atmospheric predictability: An updated perspective
,”
Mon. Weather Rev.
132
,
703
713
(
2004
).
15.
G. D.
Nastrom
and
K. S.
Gage
, “
A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft
,”
J. Atmos. Sci.
42
,
950
960
(
1985
).
16.
J. A.
Weyn
and
D. R.
Durran
, “
Ensemble spread grows more rapidly in higher-resolution simulations of deep convection
,”
J. Atmos. Sci.
75
,
3331
3345
(
2018
).
17.
T.
Selz
, “
Estimating the intrinsic limit of predictability using a stochastic convection scheme
,”
J. Atmos. Sci.
76
,
757
765
(
2019
).
18.
R.
Rotunno
and
C.
Snyder
, “
A generalization of Lorenz’s model for the predictability of flows with many scales of motion
,”
J. Atmos. Sci.
65
,
1063
1076
(
2008
).
19.
S.
Brune
and
E.
Becker
, “
Indications of stratified turbulence in a mechanistic GCM
,”
J. Atmos. Sci.
70
,
231
247
(
2013
).
20.
D. R.
Durran
and
M.
Gingrich
, “
Atmospheric predictability: Why butterflies are not of practical importance
,”
J. Atmos. Sci.
71
,
2476
2488
(
2014
).
21.
M. J. P.
Cullen
, “
The impact of high vertical resolution in the Met Office Unified Model
,”
Q. J. R. Meteorol. Soc.
143
,
278
287
(
2017
).
22.
T. Y.
Leung
,
M.
Leutbecher
,
S.
Reich
, and
T. G.
Shepherd
, “
Atmospheric predictability: Revisiting the inherent finite-time barrier
,”
J. Atmos. Sci.
76
,
3883
3892
(
2019
).
23.
Y. Q.
Sun
and
F.
Zhang
, “
A new theoretical framework for understanding multiscale atmospheric predictability
,”
J. Atmos. Sci.
77
,
2297
2309
(
2020
).
24.
J. H.
Curry
, “
A generalized Lorenz system
,”
Commun. Math. Phys.
60
,
193
204
(
1978
).
25.
P.
Reiterer
,
C.
Lainscsek
,
F.
Schürrer
,
C.
Letellier
, and
J.
Maquet
, “
A nine-dimensional Lorenz system to study high-dimensional chaos
,”
J. Phys. A: Math. Gen.
31
,
7121
7139
(
1998
).
26.
B.-W.
Shen
, “
Nonlinear feedback in a five-dimensional Lorenz model
,”
J. Atmos. Sci.
71
,
1701
1723
(
2014
).
27.
D.
Adamec
, “
Predictability of quasi-geostrophic ocean flow: Sensitivity to varying model vertical resolution
,”
J. Phys. Oceanogr.
19
,
1753
1764
(
1989
).
28.
J.
Harlim
,
M.
Oczkowski
,
J. A.
Yorke
,
E.
Kalnay
, and
B. R.
Hunt
, “
Convex error growth patterns in a global weather model
,”
Phys. Rev. Lett.
94
,
228501
(
2005
).
29.
F.
Judt
, “
Insights into atmospheric predictability through global convection-permitting model simulations
,”
J. Atmos. Sci.
75
,
1477
1497
(
2018
).
30.
J. M.
Lewis
, “
Roots of ensemble forecasting
,”
Mon. Weather Rev.
133
,
1865
1885
(
2005
).
31.
S.
Yoden
, “
Atmospheric predictability
,”
J. Meteorol. Soc. Jpn.
85B
,
77
102
(
2007
).
32.
F.
Zhang
,
C.
Snyder
, and
R.
Rotunno
, “
Effects of moist convection on mesoscale predictability
,”
J. Atmos. Sci.
60
,
1173
1185
(
2003
).
33.
R.
Buizza
, “
Horizontal resolution impact on short- and long-range forecast error
,”
Q. J. R. Meteorol. Soc.
136
,
1020
1035
(
2010
).
34.
C.
Hohenegger
and
C.
Schär
, “
Atmospheric predictability at synoptic versus cloud-resolving scales
,”
Bull. Am. Meteorol. Soc.
88
,
1783
1794
(
2007
).
35.
R.
Mureau
,
F.
Molteni
, and
T. N.
Palmer
, “
Ensemble prediction using dynamically conditioned perturbations
,”
Q. J. R. Meteorol. Soc.
119
,
299
323
(
1993
).
36.
Z.
Toth
and
E.
Kalnay
, “
Ensemble forecasting at NCEP and the breeding method
,”
Mon. Weather Rev.
125
,
3297
3319
(
1997
).
37.
D. J.
Stensrud
,
J.-W.
Bao
, and
T. T.
Warner
, “
Using initial condition and model physics perturbations in short-range ensemble simulations of mesoscale convective systems
,”
Mon. Weather Rev.
128
,
2077
2107
(
2000
).
38.
A.
Atencia
and
I.
Zawadzki
, “
Analogs on the Lorenz attractor and ensemble spread
,”
Mon. Weather Rev.
145
,
1381
1400
(
2017
).
39.
W. C.
Skamarock
and
S.-H.
Park
, “
Vertical resolution requirements in atmospheric simulation
,”
Mon. Weather Rev.
147
,
2641
2656
(
2019
).
40.
A. A.
Scaife
,
J.
Camp
,
R.
Comer
,
P.
Davis
,
N.
Dunstone
,
M.
Gordon
,
C.
MacLachlan
,
N.
Martin
,
Y.
Nie
,
H.-L.
Ren
,
M.
Roberts
,
W.
Robinson
,
D.
Smith
, and
P. L.
Vidale
, “
Does increased atmospheric resolution improve seasonal climate predictions?
,”
Atmos. Sci. Lett.
20
,
e922
(
2019
).
41.
M. L.
Waite
, “
Dependence of model energy spectra on vertical resolution
,”
Mon. Weather Rev.
144
,
1407
1421
(
2016
).
42.
G. A.
Meehl
,
L.
Goddard
,
G.
Boer
,
R.
Burgman
,
G.
Branstator
,
C.
Cassou
,
S.
Corti
,
G.
Danabasoglu
,
F.
Doublas-Reyes
,
E.
Hawkins
,
A.
Karspeck
,
M.
Kimoto
,
A.
Kumar
,
D.
Matei
,
J.
Mignot
,
R.
Msadek
,
A.
Navarra
,
H.
Phlmann
,
M.
Rienecker
,
T.
Rosati
,
E.
Schneider
,
D.
Smith
,
R.
Sutton
,
H.
Teng
,
G. J.
Van Oldenborgh
,
G.
Vecchi
, and
S.
Yeager
, “
Decadal climate prediction: An update from the trenches
,”
Bull. Am. Meteorol. Soc.
95
,
243
267
(
2021
).
43.
Y. Q.
Sun
and
F.
Zhang
, “
Intrinsic versus practical limits of atmospheric predictability and the significance of the butterfly effect
,”
J. Atmos. Sci.
73
,
1419
1438
(
2016
).
44.
E. A.
Aligo
and
W. A.
Gallus
, Jr.
, “
On the impact of WRF model vertical grid resolution on midwest summer rainfall forecasts
,”
Weather Forecast.
24
,
575
594
(
2009
).
45.
B. C.
Ancell
and
C. F.
Mass
, “
Structure, growth rates, and tangent linear accuracy of adjoint sensitivities with respect to horizontal and vertical resolution
,”
Mon. Weather Rev.
134
,
2971
2988
(
2006
).
46.
D.-L.
Zhang
and
X.
Wang
, “
Dependence of hurricane intensity and structures on vertical resolution and time-step size
,”
Adv. Atmos. Sci.
20
,
711
725
(
2003
).
47.
S. K.
Kimball
and
F. C.
Dougherty
, “
The sensitivity of idealized hurricane structure and development to the distribution of vertical levels in MM5
,”
Mon. Weather Rev.
134
,
1987
2008
(
2006
).
48.
R. S.
Lindzen
and
M.
Fox-Rabinovitz
, “
Consistent vertical and horizontal resolution
,”
Mon. Weather Rev.
117
,
2575
2583
(
1989
).
49.
M. J.
Pecnick
and
D.
Keyser
, “
The effect of spatial resolution on the simulation of upper-tropospheric frontogenesis using a sigma-coordinate primitive equation model
,”
Meteorol. Atmos. Phys.
40
,
137
149
(
1989
).
50.
P. G.
Persson
and
T. T.
Warner
, “
Model generation of spurious gravity waves due to inconsistency of the vertical and horizontal resolution
,”
Mon. Weather Rev.
119
,
917
935
(
1991
).
51.
W. C.
Skamarock
, “
Evaluating mesoscale NWP models using kinetic energy spectra
,”
Mon. Weather Rev.
132
,
3019
3032
(
2004
).
52.
L.
De Cruz
,
S.
Schubert
,
J.
Demaeyer
,
V.
Lucarini
, and
S.
Vannitsem
, “
Exploring the Lyapunov instability properties of high-dimensional atmospheric and climatic models
,”
Nonlinear Process. Geophys.
25
,
387
412
(
2018
).
53.
C.
Stegemann
,
H. A.
Albuquerque
,
R. M.
Rubinger
, and
P. C.
Rech
, “
Lyapunov exponent diagrams of a 4-dimensional Chua system
,”
Chaos
21
,
033105
(
2011
).
54.
E. N.
Lorenz
, “Predictability—A problem partly solved,” in Predictability of Weather and Climate, edited by T. Palmer and R. Hagedorn (Cambridge University Press, Cambridge, 2006), Chap. 3, pp. 40–58.
55.
T.
Bódai
, “
Predictability of threshold exceedances in dynamical systems
,”
Physica D
313
,
37
50
(
2015
).
56.
X.
Zeng
,
R.
Eykholt
, and
R. A.
Pielke
, “
Estimating the Lyapunov-exponent spectrum from short time series of low precision
,”
Phys. Rev. Lett.
66
,
3229
3232
(
1991
).
57.
W. C.
Skamarock
,
J. B.
Klemp
,
J.
Dudhia
,
D. O.
Gill
,
Z.
Liu
,
J.
Berner
,
W.
Wang
,
J. G.
Powers
,
M. G.
Duda
,
D. M.
Barker
, and
X.-Y.
Huang
, “A description of the advanced research WRF version 4,” NCAR Technical Notes NCAR/TN-556+STR (2019), p. 145.
58.
N.
Bei
and
F.
Zhang
, “
Impacts of initial condition errors on mesoscale predictability of heavy precipitation along the Mei-Yu front of China
,”
Q. J. R. Meteorol. Soc.
133
,
83
99
(
2007
).
59.
E.
Lee
,
E.-H.
Lee
, and
I.-J.
Choi
, “
Impact of increased vertical resolution on medium-range forecasts in a global atmospheric model
,”
Mon. Weather Rev.
147
,
4091
4106
(
2019
).
60.
See https://www.ecmwf.int/en/forecasts/documentation-and-support for ECMWF, “L137 Model Level Definitions” (2021); accessed 24 March 2021.
61.
See https://cds.climate.copernicus.eu/cdsapp#!/home for Copernicus Climate Change Service (C3S), “ERA5: Fifth Generation of ECMWF Atmospheric Reanalysis of the Global Climate” (Copernicus Climate Change Service Climate Data Store (CDS), 2017); accessed 20 December 2019.
62.
C.
Hohenegger
and
C.
Schär
, “
Predictability and error growth dynamics in cloud-resolving models
,”
J. Atmos. Sci.
64
,
4467
4478
(
2007
).
63.
J.
Blázquez
,
N. L.
Pessacg
, and
P. L. M.
Gonzalez
, “
Simulation of a baroclinic wave with the WRF regional model: Sensitivity to the initial conditions in an ideal and a real experiment
,”
Meteorol. Appl.
20
,
447
456
(
2013
).
64.
L. S. R.
Froude
,
L.
Bengtsson
, and
K. I.
Hodges
, “
Atmospheric predictability revisited
,”
Tellus
65A
,
19022
(
2013
).
65.
P.
Marquet
,
J.-F.
Mahfouf
, and
D.
Holdaway
, “
Definition of the moist-air exergy norm: A comparison with existing “moist energy norms”
,”
Mon. Weather Rev.
148
,
907
928
(
2020
).
66.
M.
Borderies
,
O.
Caumont
,
J.
Delanoë
,
V.
Ducrocq
,
N.
Fourrié
, and
P.
Marquet
, “
Impact of airborne cloud radar reflectivity data assimilation on kilometre-scale numerical weather prediction analyses and forecasts of heavy precipitation events
,”
Nat. Hazards Earth Syst. Sci.
19
,
907
926
(
2019
).
67.
G. J.
Huffman
,
E. F.
Stocker
,
D. T.
Bolvin
,
E. J.
Nelkin
, and
J.
Tan
, “GPM IMERG Final Precipitation L3 Half Hourly 0.1 Degree × 0.1 Degree V06” (Goddard Earth Sciences Data and Information Services Center (GES DISC), Greenbelt, MD, 2019); accessed 29 January 2021.
68.
E. N.
Lorenz
, “
Atmospheric predictability experiments with a large numerical model
,”
Tellus
34
,
505
513
(
1982
).
69.
J. T.
Schaefer
, “
The critical success index as an indicator of warning skill
,”
Weather Forecast.
5
,
570
575
(
1990
).
70.
C.-C.
Wang
, “
On the calculation and correction of equitable threat score for model quantitative precipitation forecasts for small verification areas: The example of Taiwan
,”
Weather Forecast.
29
,
788
798
(
2014
).
71.
C. M.
Wheeler
,
H.
Zu
,
A. H.
Sobel
,
D.
Hudson
, and
F.
Vitart
, “
Seamless precipitation prediction skill comparison between two global models
,”
Q. J. R. Meteorol. Soc.
143
,
374
383
(
2017
).
72.
A. A.
Tsonis
and
J. B.
Elsner
, “
Chaos, strange attractors, and weather
,”
Bull. Am. Meteorol. Soc.
70
,
14
23
(
1989
).
73.
D.
Schertzer
and
S.
Lovejoy
, “Uncertainty and predictability in geophysics: Chaos and multifractal insights,” in The State of the Planet: Frontiers and Challenges in Geophysics, Geophysical Monograph Series Vol. 150, edited by R. S. J. Sparks and C. J. Hawkesworth (American Geophysical Union, Washington, DC, 2004), pp. 317–334.
74.
R.
Buizza
, “
Introduction to the special issue on “25 years of ensemble forecasting”
,”
Q. J. R. Meteorol. Soc.
145
,
1
11
(
2019
).
75.
D. A.
Lavers
,
F.
Pappenberger
, and
E.
Zsoter
, “
Extending medium-range predictability of extreme hydrological events in Europe
,”
Nat. Commun.
5
,
5382
(
2019
).
76.
J.
Park
,
S.
Moon
,
J. M.
Seo
, and
J.-J.
Baik
, “
Systematic comparison between the generalized Lorenz equations and DNS in the two-dimensional Rayleigh–Bénard convection
,”
Chaos
31
,
073119
(
2021
).
77.
N.
On
,
H. M.
Kim
, and
S.
Kim
, “
Effects of resolution, cumulus parameterization scheme, and probability forecasting on precipitation forecasts in a high-resolution limited-area ensemble prediction system
,”
Asia-Pac. J. Atmos. Sci.
54
,
623
637
(
2018
).
78.
S.-Y.
Hong
,
Y.
Noh
, and
J.
Dudhia
, “
A new vertical diffusion package with an explicit treatment of entrainment processes
,”
Mon. Weather Rev.
134
,
2318
2341
(
2006
).
79.
J.
Dudhia
, “
Numerical study of convection observed during the winter monsoon experiment using a mesoscale two-dimensional model
,”
J. Atmos. Sci.
46
,
3077
3107
(
1989
).
80.
E.
Mlawer
,
S. J.
Taubman
,
P. D.
Brown
,
M. J.
Iacono
, and
S. A.
Clough
, “
Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k model for the longwave
,”
J. Geophys. Res.
102
,
16663
16682
, https://doi.org/10.1029/97JD00237 (
1997
).
81.
M.
Tewari
,
F.
Chen
,
W.
Wang
,
J.
Dudhia
,
M. A.
LeMone
,
K.
Mitchell
,
M.
Ek
,
G.
Gayno
,
J.
Wegiel
, and
R. H.
Cuenca
, “Implementation and verification of the unified NOAH land surface model in the WRF model,” in 20th Conference on Weather Analysis and Forecasting/16th Conference on Numerical Weather Prediction (American Meteorological Society, 2004), pp. 11–15.
82.
J. S.
Kain
, “
The Kain–Fritsch convective parameterization: An update
,”
J. Appl. Meteorol.
43
,
170
181
(
2004
).
83.
S.-Y.
Hong
and
J.-O. J.
Lim
, “
The WRF single-moment 6-class microphysics scheme (WSM6)
,”
J. Kor. Meteorol. Soc.
42
,
129
151
(
2006
).
You do not currently have access to this content.