Nonlinear anticyclone structures in the Earth's atmosphere
We present some results of the study of nonlinear vortex structures in the Earth's atmosphere known as blocking-anticyclones. It is shown that synoptic-scale structure can be considered in the geostrophic approximation in terms of the Charney-Obukhov equation. The numerical scheme based on the...
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Advances in astronomy and space physics
2011
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| Цитувати: | Nonlinear anticyclone structures in the Earth's atmosphere / D. Saliuk, O. Agapitov // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 69-72. — Бібліогр.: 5 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1189732025-02-09T22:52:22Z Nonlinear anticyclone structures in the Earth's atmosphere Saliuk, D. Agapitov, O. We present some results of the study of nonlinear vortex structures in the Earth's atmosphere known as blocking-anticyclones. It is shown that synoptic-scale structure can be considered in the geostrophic approximation in terms of the Charney-Obukhov equation. The numerical scheme based on the full reduction method is proposed to study the dynamics of the initial perturbations. The dominant role of the vector nonlinearity for the vortex structures stability is shown. In the absence of the nonlinearity the vortex structures are unstable and quickly decompose into the linear Rossby waves. The linear and nonlinear anticyclone structures dynamics in the Earth's atmosphere were studied using the data from NCEP(National Centers for Environmental Prediction) / NCAR (National Center for Atmospheric Research) data base - pressure, geopotential, temperature, wind velocity. Blocking anticyclones exist during a long time (typical lifetime is from 5 days to a month), while linear anticyclone exist few days (up to 5 days). Conventional synoptic anticyclone usually moves eastward but blocking anticyclones usually move westward (or stay on the same place in the zonal flow). It was also found that blocking anticyclones have larger amplitude of temperature and pressure perturbations, so the impact of the blocking anticyclone on the formation of weather is larger. We would like to thank Atmospheric Research NCEP / NCAR data service for atmospheric data. 2011 Article Nonlinear anticyclone structures in the Earth's atmosphere / D. Saliuk, O. Agapitov // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 69-72. — Бібліогр.: 5 назв. — англ. 987-966-439-367-3 https://nasplib.isofts.kiev.ua/handle/123456789/118973 en Advances in Astronomy and Space Physics application/pdf Advances in astronomy and space physics |
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We present some results of the study of nonlinear vortex structures in the Earth's atmosphere known as blocking-anticyclones. It is shown that synoptic-scale structure can be considered in the geostrophic approximation in terms of the Charney-Obukhov equation. The numerical scheme based on the full reduction method is proposed to study the dynamics of the initial perturbations. The dominant role of the vector nonlinearity for the vortex structures stability is shown. In the absence of the nonlinearity the vortex structures are unstable and quickly decompose into the linear Rossby waves. The linear and nonlinear anticyclone structures dynamics in the Earth's atmosphere were studied using the data from NCEP(National Centers for Environmental Prediction) / NCAR (National Center for Atmospheric Research) data base - pressure, geopotential, temperature, wind velocity. Blocking anticyclones exist during a long time (typical lifetime is from 5 days to a month), while linear anticyclone exist few days (up to 5 days). Conventional synoptic anticyclone usually moves eastward but blocking anticyclones usually move westward (or stay on the same place in the zonal flow). It was also found that blocking anticyclones have larger amplitude of temperature and pressure perturbations, so the impact of the blocking anticyclone on the formation of weather is larger. |
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Article |
| author |
Saliuk, D. Agapitov, O. |
| spellingShingle |
Saliuk, D. Agapitov, O. Nonlinear anticyclone structures in the Earth's atmosphere Advances in Astronomy and Space Physics |
| author_facet |
Saliuk, D. Agapitov, O. |
| author_sort |
Saliuk, D. |
| title |
Nonlinear anticyclone structures in the Earth's atmosphere |
| title_short |
Nonlinear anticyclone structures in the Earth's atmosphere |
| title_full |
Nonlinear anticyclone structures in the Earth's atmosphere |
| title_fullStr |
Nonlinear anticyclone structures in the Earth's atmosphere |
| title_full_unstemmed |
Nonlinear anticyclone structures in the Earth's atmosphere |
| title_sort |
nonlinear anticyclone structures in the earth's atmosphere |
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Advances in astronomy and space physics |
| publishDate |
2011 |
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https://nasplib.isofts.kiev.ua/handle/123456789/118973 |
| citation_txt |
Nonlinear anticyclone structures in the Earth's atmosphere / D. Saliuk, O. Agapitov // Advances in Astronomy and Space Physics. — 2011. — Т. 1., вип. 1-2. — С. 69-72. — Бібліогр.: 5 назв. — англ. |
| series |
Advances in Astronomy and Space Physics |
| work_keys_str_mv |
AT saliukd nonlinearanticyclonestructuresintheearthsatmosphere AT agapitovo nonlinearanticyclonestructuresintheearthsatmosphere |
| first_indexed |
2025-12-01T13:55:21Z |
| last_indexed |
2025-12-01T13:55:21Z |
| _version_ |
1850314403759521792 |
| fulltext |
Nonlinear anticyclone structures in the Earth's atmosphere
D. Saliuk, O. Agapitov
Taras Shevchenko National University of Kyiv, Glushkova ave., 4, 03127, Kyiv, Ukraine
dima.ubf@gmail.com
We present some results of the study of nonlinear vortex structures in the Earth's atmosphere known
as blocking-anticyclones. It is shown that synoptic-scale structure can be considered in the geostrophic
approximation in terms of the Charney-Obukhov equation. The numerical scheme based on the full reduction
method is proposed to study the dynamics of the initial perturbations. The dominant role of the vector
nonlinearity for the vortex structures stability is shown. In the absence of the nonlinearity the vortex
structures are unstable and quickly decompose into the linear Rossby waves. The linear and nonlinear
anticyclone structures dynamics in the Earth's atmosphere were studied using the data from NCEP(National
Centers for Environmental Prediction) / NCAR (National Center for Atmospheric Research) data base �
pressure, geopotential, temperature, wind velocity. Blocking anticyclones exist during a long time (typical
lifetime is from 5 days to a month), while linear anticyclone exist few days (up to 5 days). Conventional
synoptic anticyclone usually moves eastward but blocking anticyclones usually move westward (or stay on
the same place in the zonal �ow). It was also found that blocking anticyclones have larger amplitude of
temperature and pressure perturbations, so the impact of the blocking anticyclone on the formation of
weather is larger.
Introduction
Dynamics of the waves in the atmosphere of a rotating planet can be described with the following system
of equations {
dV
dt = −∇ (gH) + Ω [V, ζ] ,
∂H
∂t = div (VH) ,
(1)
where V is the horizontal component of the �ow velocity in the atmosphere, ζ is the vector normal to the
planet's surface, Ω is the planet's rotation frequency, g is the acceleration of gravity near the planet surface,
H is the pressure scale height, which depends on the sea level pressure (for the atmosphere with molecules
of e�ective masses M and temperatures T , undisturbed pressure scale height is H0 = kBT
Mg , where kB is the
Boltzmann constant; for Earth H0 is about 8 km, for Jupiter H0 is about 25 km). Taking into account the
approximation for the atmosphere as an incompressible layer of liquid with depth equal to H one can obtain
dispersion equation for oscillations with small amplitudes
ω
(
1 + k2r2
R −
ω2
Ω2
)
= −kϕV∗, (2)
where rR = (gH0)
1
2
Ω is the barotropic Rossby radius (it is about 2000 km for the Earth and 6000 km for
Jupiter), where k is the wave vector, V∗ is the Rossby velocity, kϕ is the longitudinal projection of the wave
vector. For frequencies much less then Ω the dispersion equation for Rossby waves can be found as:
ω = − kϕV∗
1 + k2r2
R
.
Thus linear barotropic Rossby waves have the westward phase velocity of [3, 5].
Vph = − V∗
1 + k2r2
R
.
69
Advances in Astronomy and Space Physics D. Salyuk, O. Agapitov
When the wavelength tends to in�nity, the phase velocity tends to the Rossby velocity. Taking into
account Ertel theorem one can decompose Eq. (1) in the small parameter and obtain [4]:
∂
∂t
(
h− r2
R∆h
)− V∗
R
∂h
∂y
(
h +
h2
2
)
= Ωr4
R [∇h,∇∆h] , (3)
where h = H−H0
H . Then, using well known β-plane approximation, we introduce the two-dimensional local
Cartesian system of coordinates (x, y) with longitude and latitude being x and y correspondingly. x0 and
y0 are longitude and latitude of the plane contact point. In this coordinate system the x-axis is directed
from the west to the east and the y-axis points to the north. Then with the β-plane approximation we get
non-dimensional Charney-Obukhov equation
∂(∆h− h)
∂t
+ β
∂h
∂y
+ h
∂h
∂y
+ {h,∆⊥h} = 0, (4)
where V∗ = gH0
2ω0R sin2 a
is the non-dimensional Rossby velocity and {h,∆⊥h} = ∂xh∂y∆h− ∂yh∂x∆h denotes
the Poisson bracket. The second and third parts of Eq. (4) include two types of nonlinearities: the scalar
and the vector one. The scalar nonlinearity is directly related to the changes of the thickness of H layer.
It is usually included in the equation for nonlinear waves, such as the Korteweg�de Vries equation obtained
from �nite amplitude waves on the shallow water � the �rst soliton in the history of science, observed by
Scott Russel about 170 years ago. Vector nonlinearity may not be related to changes of H. Two kinds of
nonlinearities may be subdivided only asymptotically. Scalar nonlinearity vanishes in the absence of liquid
free surface, while the vector nonlinearity disappears under two conditions: axial symmetry and the Rossby
velocity independence on latitude. The equation (4) has a form similar to the Hasegawa-Mina equation
for an inhomogeneous plasma [2]. Taking the ratio of the third term to the fourth one one can obtain the
following relation [4, 5]:
scalar
vector
≈ a2
r2
R
. (5)
In the case of a > rR the scalar nonlinearity dominates, while for a < rR the vector nonlinearity dominates.
The ratio (5) are estimative and useful for experimental search of the conditions under which we can expect
the prevalence of the �rst or the second kind of nonlinearity.
From equation (4) the asymmetry of cyclones and anticyclones properties follows. The scalar nonlinearity
can be in a balance with the anticyclone dispersion according to equation (4). Cyclones have the same signs
in the dispersion and scalar nonlinearity and therefore can not be mutually compensated. This cyclone-
antycyclone asymmetry mainly determines the possibility (or impossibility) of formation of single vortices of
di�erent polarity.
Algorithm of the numerical simulations
We consider the numerical scheme for study the dynamics of the time-dependent perturbations of a system
described by equation Charney-Obukhov (or its generalization taking into account the scalar type of linearity
in a form of the Korteveg-de Vries equation):
∂ (h−∆⊥h)
∂t
− β
∂h
∂y
= {h,∆⊥h} . (6)
We used the Arakawa numerical scheme for the spatial derivatives [1]. The IDL procedure of numerical
integration algorithm on a grid size of 50× 50 and 100× 100 with increments of 0.1 in time and space was
used for simulations. The initial conditions are de�ned by a two dimensional Gaussian with characteristic
scale about Rossby radius. The nature of the solution depends on the ratio between linear and nonlinear
component in the equation. In a case of scalar component predominance the initial disturbance in a form
of the monopole vortex is unstable. It decomposes to the linear Rossby waves due to dispersion of the
drift velocity. In a case of the signi�cant nonlinear component occurrence the nonlinear rotational solution
is stable. It is also somewhat dispersed during the drift, but remains localized in space and retains the
characteristic features.
70
Advances in Astronomy and Space Physics D. Salyuk, O. Agapitov
Figure 1: Dynamics of the initial perturbation in the form of the monopoly vortex with the Gaussian potential:
dynamics of the linear system (top panel: the initial structure decomposes to the linear waves) and dynamics of
the nonlinear system (bottom panel: it retains the basic The spatial scale is in units of Rossby radius rR. Time
is in rR/v∗
.
Evolution in time of the initial conditions on the grid is presented in Fig. 1, where the atmosphere
disturbance h is shown at times moments 0, 4, 36 and 48 of rR/v∗ units. The linear case can also be used as
a test case to verify the algorithm. The results are in a good agreement with the properties of small scale
anticyclones in the Earth atmosphere described in [4, 5]. For experimental veri�cation we used the data
from National Center for Environmental Prediction (NCEP) and National Center for Atmospheric Research
(NCAR) data bases. These data are public available1 in NetCDF format. The �les contain the data with
step 2.5◦ in latitude and longitude, divided into 17 levels in height from 1000 mB to 10 mB, where each level
contains six variables that represent geopotential altitude, temperature, v-and u-wind components, and other
variables with a time step of 1 day. In general, the data has dimension [144,73,17,365] for 144 longitudes
values, 73 latitude values and 365 days.The small scale vortex (spatial scale is about rR) structure observed
on September 17, 2001 is shown in Fig. 2 in the geopotential values. The structure was observed during more
then 10 days on the same place. The background disturbances were much more dynamic with characteristic
time scale about 3�4 days.
Results and conclusions
The study of the nonlinear vortex formation in the Earth's atmosphere known as a blocking anticyclone is
proposed. It is shown that this synoptic-scale structure can be considered in the geostrophic approximation
in terms of the Charney-Obukhov equation. The numerical scheme based on the full reduction method is
proposed to study the dynamics of the initial perturbations. The dominant role of the vector nonlinearity
for the vortex structures stability is shown. In the absence of the nonlinearity the vortex structures are
unstable and quickly decompose into the linear Rossby waves. The di�erences between linear and nonlinear
anticyclone structures were studied using the data from Atmospheric Research NCEP/NCAR data base.
Blocking anticyclones exist during a long time (typical lifetime is from 5 days to a month), while linear
anticyclones exist during few days (up to 5 days). Blocking anticyclones can stay for a long time on the
same place (Fig. 2). Conventional synoptic anticyclones usually move eastward, while blocking anticyclones
1http://www.cdc.noaa.gov/cdc/data.ncep.reanalysis.pressure.html
71
Advances in Astronomy and Space Physics D. Salyuk, O. Agapitov
Figure 2: The geopotetial map that shows location of blocking anticyclone on 17.09.2001.
usually move westward. It was also found that blocking anticyclones have larger amplitude of temperature
and pressure perturbation, so the impact of the blocking anticyclone on the formation of weather is larger.
Acknowledgement
We would like to thank Atmospheric Research NCEP / NCAR data service for atmospheric data.
References
[1] Arakawa A. J. Comput. Phys., V. 135, pp. 103-114 (1997)
[2] Horton W., Hasegawa A. Chaos: An Interdisciplinary J. of Nonlinear Science, V. 4, No. 2, pp. 227-251 (1994)
[3] Monin A. S., Zhikharev G. M. Advances in Physical Sciences, V. 160, no. 5, pp. 1-48 (1990)
[4] Nezlin M.V. Advances in Physical Sciences, V. 150, no. 1, pp. 3-60 (1986)
[5] Petviashvili V. I., Pokhotelov V. A. Solitary waves in planetary atmospheres and oceans. Energoatomizdat, Moscow (1989)
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