Mathematical Modeling of the Influence of the Wind Field Structure in the Atmosphere on the Cloud Formation Processes

Full text

Atmospheric and Climate Sciences, 2018, 8, 84-96
http://www.scirp.org/journal/acs
ISSN Online: 2160-0422
ISSN Print: 2160-0414
DOI:
10.4236/acs.2018.81006 Jan. 17, 2018 84 Atmospheric and Climate Sciences
Mathematical Modeling of the Influence of the
Wind Field Structure in the Atmosphere on the
Cloud Formation Processes
Boris A. Ashabokov1,2, Lyudmila M. Fedchenko1, Alexander V. Shapovalov1,
Khazhbara M. Kalov1, Ruslan Kh. Kalov1, Alla A. Tashilova1, Vitaly A. Shapovalov1
1High-Mountain Geophysical Institute, Nalchik, Russia
2Institute of Computer Science and Problems of Regional Management, Kabardino-Balkarian Research Center,
Russian Academy of Sciences, Nalchik, Russia
Abstract
The state of the physics of convective clouds and cloud seeding is discussed
briefly. It is noted that at the present time there is a transition from the stage
of investigation of “elementary” processes in the clouds to the stage of stud
y-
ing the formation of macro-
and microstructural characteristics of clouds as a
whole, taking into account their system properties. The main directions of the
development of cloud physics at the upcoming stage of
its development are
discussed. The paper points out that one of these areas is the determination of
the structure-
forming factors for the clouds and the study of their influence
on their formation and evolution. It is noted that one of such factors is the i
n-
teraction of clouds with their surrounding atmosphere, and the main method
of studying its role in the processes of cloud formation is mathematical mo
d-
eling. A three-dimensional nonstationary model of convective clouds is pr
e-
sented with a detailed accou
nt of the processes of thermohydrodynamics and
microphysics, which is used for research. The results of modeling the infl
u-
ence of the wind field structure in the atmosphere on the formation and ev
o-
lution of clouds are presented. It is shown that the dynami
c characteristics of
the atmosphere have a significant effect on the formation of macro- and m
i-
crostructural characteristics of convective clouds: the more complex the
structure of the wind field in the atmosphere (
i.e.
, the more intense the int
e-
raction of
the atmosphere and the cloud), the less powerful the clouds are
formed.
Keywords
Wind in the Atmosphere, Wind Field Structure, Interaction with Clouds,
Modeling, System Properties of Clouds, Interaction of Clouds with the
How to cite this paper:
Ashabokov, B.A.,
Fedchenko
, L.M., Shapovalov, A.V., Kalov
,
K
.M., Kalov, R.K., Tashilova, A.A. and Sha-
povalov
, V.A. (2018) Mathematical Model
ing
of the Influence of the Wind Field
Structure in
the Atmosphere on the Cloud Formation
Processes
.
Atmospheric and Climate Sciences
,
8
, 84-96.
https://doi.org/10.4236/acs.2018.81006
Received:
December 14, 2017
Accepted:
January 14, 2018
Published:
January 17, 2018
Copyright © 201
8 by authors and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access

B. A. Ashabokov et al.
DOI:
10.4236/acs.2018.81006 85 Atmospheric and Climate Sciences
Atmosphere, Three-Dimensional Cloud Model
1. Introduction
The present period of time is a transition period for cloud physics and cloud
seeding: a gradual transition from the stage of studying the “elementary”
processes in the clouds to the stage of studying the formation and evolution of
clouds as a whole, taking into account their system properties [1] [2]. Obviously,
such a transition is natural, because cloud physics is not limited to the study of
individual processes in the clouds, and there are still many factors affecting the
formation and evolution of clouds and requiring study. It should also be noted
that the transition of cloud physics to the next stage of evolution will take a cer-
tain period of time. This is due to the fact that, on the one hand, such “elemen-
tary” processes, whose role in the formation and evolution of clouds are great,
remained unexplored or insufficiently studied, and, on the other hand, it is ne-
cessary to formulate research directions at the next stage of the evolution of
cloud physics, the tasks of these directions, develop methodologies and methods
for solving these problems. Taking into account that convective clouds belong to
complex physical systems, one of the directions of research of a new stage of its
evolution is the determination of the main structure-forming factors for clouds
and the study of their influence on the formation of their macro- and micro-
structural characteristics. Such factors, as noted in [1] [2], are the interaction of
clouds with their surrounding atmosphere and the interaction of processes in the
clouds (the properties of hierarchy and emergence of systems) [3] [4].
In this paper, we present some results of studies on the effect of the structure
of the wind field in the atmosphere on the formation of macro- and microstruc-
tural characteristics of convective clouds. It is one of the most important me-
chanisms of interaction of clouds with their surrounding atmosphere.
Note that this problem has long been in the field of view of researchers. As an
example, we can mention papers [5]-[11], which are devoted to the investigation
of the possible influence on the processes of cloud formation of various mechan-
isms of interaction of the atmosphere with clouds. The list of these works can be
continued, but without dwelling on this, we note that the authors of these works
did not distinguish the belonging of this problem to a new stage in the evolution
of cloud physics. In addition, the possibilities of the methods that they used were
significantly limited. Therefore, these studies should be carried out from a new
perspective and using more effective methods and methodologies.
In this regard, it is important to note that the study of the influence of this
factor on the processes of cloud- and precipitation formation processes is possi-
ble only on the basis of numerical modeling using full three-dimensional cloud
models. In this paper we use for this purpose a three-dimensional numerical
model of mixed convective clouds with a detailed account of the processes [1]
[2] [12] [13]. The model is effectively used to study various issues related to the

B. A. Ashabokov et al.
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10.4236/acs.2018.81006 86 Atmospheric and Climate Sciences
processes of cloud formation. In particular, on the basis of this model, the for-
mation of the thermodynamic, microstructural and electrical characteristics of
convective clouds was studied [14] [15] [16], and the formation of their micro-
structural characteristics was studied in [17] [18]. Of the works devoted to the
study of the role of the interaction of processes in the clouds on the formation of
their microstructural characteristics, it can be noted [19]. It presents the results
of modeling the influence of deformations in the clouds (formation of the inte-
raction of processes in the clouds) on the formation of microstructural characte-
ristics of hail clouds. We also note that a new method based on the use of the
“digital atmosphere” is used to form the input data of the model (initial condi-
tions) [20].
2. Research Method
The present work uses a three-dimensional model with detailed account of
processes, including electrical ones, to study the effect of the wind field structure
in the atmosphere on the formation of macro- and microstructural characteris-
tics of convective clouds.
We present a system of equations describing processes in the clouds to get an
idea of the model, and for a more detailed acquaintance with the model, we can
recommend the papers [4] [5] [6].
The mathematical model of the convective cloud includes the equations of
thermodynamics, microphysics and electrostatics:
( )
uu u lv
t
π
∂′′
+ ⋅∇ = −∇ + ∆ +
∂V
, (1)
( )
vv v lu
t
π
∂′′
+ ⋅∇ = −∇ + ∆ −
∂V
, (2)
( )
0
0,61
S
w
w wg sQ
t
θ
πθ

′
∂′′ ′
+ ⋅∇ = −∇ + ∆ + + −

∂
V
, (3)
Continuity equation:
uw
w
xyz
νσ
∂∂∂
++=
∂∂∂
, (4)
Equations of thermodynamics:
( )
С С З З
К К
p pp
L ML M
LM
t cT t cT t cT t
δδ
δ
θ θ θθ
θθ
δ δδ
∂′
+ ⋅∇ = + + +∆
∂V
, (5)
( )
C
К
M
M
sss
t tt
δ
δ
δδ
∂′
+ ⋅∇ =− − +∆
∂V
, (6)
Equations for the distribution functions of droplets, crystals, and
frost-fragmentation by mass:
( )
111 1
1
11 1 1 1
11
КKГ АК ДР З
fff f
u v wV
txy z
ff f f f
fI
tt t t t
∂∂∂ ∂
+ + +−
∂∂∂ ∂
∂∂ ∂ ∂ ∂
    ′
= + + + + +∆ +
   
∂∂ ∂ ∂ ∂
   
, (7)

B. A. Ashabokov et al.
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10.4236/acs.2018.81006 87 Atmospheric and Climate Sciences
( )
222 2
2
22 2
22АВ
CAК З
fff f
u v wV
txy z
ff f
fII
tt t
∂∂∂ ∂
+ + +−
∂∂∂ ∂
∂∂ ∂
  ′
= + + +∆ + +
 
∂∂ ∂
 
, (8)
( )
333 33 3
23
З АК
fff ff f
u v wV f
txy zt t
∂∂∂ ∂∂ ∂
 ′
+ + + − = + +∆

∂∂∂ ∂∂ ∂

, (9)
Poisson's equation for the potential of an electrostatic field:
222 e
222 0
UUU
xyz
ρ
ε
∂∂∂
++=−
∂∂∂
. (10)
The initial conditions for Equations (1)-(12) have the following form:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
00 0
00
,0 , ,0 , ,0 ,
,0 , ,0 ,
u uv vw w
ss
θθ
= = =
= =
r rr r r r
r rr r
(11)
( ) ( ) ( ) ( ) ( )
123
, ,0 , ,0 , ,0 0, ,0 ,0 0.fmfmfm
ρρ
−+
= = = = =r r r rr
(12)
Border conditions:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
00 0
00
0, ; 0, ;
, , , , , ,
, , ,
x yz
x L y L zL
utu vt v wt w
t sts
θθ
= = =
= = =
= =
r rr r r r
r rr r
() () ( ) ( ) ( ) ( ) ( )
00
0
, , , 0, , , ,
z
utvtwt t sts
θθ
=
= = = = =r r r r rr r
(13)
( ) ( ) ( )
123 0, ; 0, ;
,, ,, ,, 0 x yz
x L y L zL
f mt f mt f mt
= = =
= = =rrr
()() ( )
123
0
,, ,, ,, 0
z
f mt f mt f mt
zzz
=
∂∂∂
= = =
∂∂∂
rrr
(14)
( ) ( ) ( ) ( )
0
0, 0,
,,,
0,0,0,,0
x yz
z
x L y L zL
Ut Ut Ut Ut
xyz
=
= = =
∂∂∂
= = = =
∂∂∂
rrr
r
(15)
The system of equations is applied for the space-time domain:
0 ,0 ,0 ,0 , 0
x yz
xL yL zL m t≤ ≤ ≤ ≤ ≤ ≤ ≤ <∞ >
. (16)
The following notation is used:
( )
,uvw K K K
x y z xxyyzz
∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂
′
⋅∇ ≡ + + ∆ = + +
∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂
V
,
{ }
,,xyz=r
-coordinate vector,
{ }
,,uvw=V
-velocity vector,
( )
ur
,
( )
vr
,
( )
wr
-velocity vector components;
l
-inertial force parameter;
( )
θ
r
-potential
temperature;
( ) ( )
( )
1000
p
RC
p
c Pz
πθ
=r
-dimensionless pressure;
θ
-average
potential temperature;
R
-gas constant;
( )
sr
-specific air humidity;
( )
S
Qr
-the
total ratio of the mixture of liquid and solid phases in the cloud;
( )
z
σ
-a para-
meter that takes into account the change in air density with altitude;
( )
Pz
и
( )
Tr
-respectively, pressure and temperature;
p
c
-heat capacity of air at con-
stant pressure;
,,
KCЗ
LLL
-respectively, the specific heat of condensation, subli-
mation and freezing;
( ) ( ) ( )
,,s
πθ
′′′
rrr
-deviations of dimensionless pressure,
potential temperature and specific humidity from their background values in the

B. A. Ashabokov et al.
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10.4236/acs.2018.81006 88 Atmospheric and Climate Sciences
ambient atmosphere
()( ) ( )
000
,,
s
πθ
rrr
;
K
М
t
δ
δ
,
C
М
t
δ
δ
-changes in specific
humidity due to the diffusion of steam into droplets and crystals;
З
М
t
δ
δ
-the
mass of the dropping water that freezes per unit time in a unit volume of air;
()
Kr
-turbulent diffusion coefficient.
( ) ( )
12
,VmV m
-steady rate of fall of liquid and
solid particles;
1
К
ft
∂


∂

,
1
КГ
ft
∂


∂

,
1
АК
ft
∂


∂

,
1
ДР
ft
∂


∂

,
1
З
ft
∂


∂

-changes in the
distribution function of droplets due to microphysical condensation processes,
coagulation of droplets, accretion of droplets and crystals, crushing and freezing,
respectively;
2
C
ft
∂


∂

,
2
АК
ft
∂


∂

,
2
З
ft
∂


∂

-changes in the distribution function of
crystals due to sublimation, accretion and freezing of droplets;
3
З
ft
∂


∂

,
3
АК
ft
∂


∂

-changes in the distribution function
( )
3,,f mtr
due to the formation
of fragments during spontaneous freezing of supercooled cloud droplets and
their accretion with crystals;
12
and
II
-sources of droplets and crystals;
АВ
I
-source of artificial crystals under cloud seeding;
0
ε
-dielectric constant of va-
cuum.
The boundaries of the spatial domain are denoted by
0, ,0,
xy
LL
и
0,
z
L
The remaining notations are given in the works [4] [5] [6].
Powerful convective clouds observed in the North Caucasus on 02.09.2010
and 07.06.2012 and accompanied by the fall of hail were selected for the study.
Calculations were made of the formation and evolution of clouds for various
wind field structures in the atmosphere, which included the actual structure
constructed from the aerological sounding data (the Mineralnye Vody Airport,
North Caucasus, Russia) of the atmosphere on the days indicated, and the model
wind structures constructed using the same data but modified as planned au-
thors. The remaining parameters of the atmosphere remained unchanged during
the calculations. In Figure 1, the real (left) and model (right) wind field structures
in the atmosphere are given as an example (02.06.2010). The difference between
the real and model field structures, as you can see, is that in the second case, at all
levels, the wind velocity vector in the atmosphere is directed along the axis OX.
Comparative analysis of the characteristics of the model clouds (
i.e.
based on
the calculation results) to study the effect of the wind field structure on the
processes of cloud- and precipitation formation processes at the same time in-
stants was carried out: the characteristics of the velocity fields of vertical air
movements in the cloud and near cloudiness, radar reflectivity, as well as the
maximum values of cloud parameters.
3. Results of Calculations
Let us dwell on the results of calculations of the cloud corresponding to the real
structure of the wind field in the atmosphere. In the Figure 2 and Figure 3

B. A. Ashabokov et al.
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Figure 1. Real (left) and model (right) structure of the wind field in the atmosphere, used
for calculations.
Figure 2. Isolines of upflow velocity in the vertical plane (20th min).
Figure 3. Isolines of different values of the upflow velocity in the vertical plane (30th
min).

B. A. Ashabokov et al.
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10.4236/acs.2018.81006 90 Atmospheric and Climate Sciences
isolines of vertical velocities in the cloud and near cloud are shown in a vertical
plane passing through the center of the cloud along the axis OX. The figures
correspond to 20 and 30 min of the evolution of the cloud. It can be noted that
the velocity field of the ascending and descending air currents has a rather com-
plex structure, and, as can be seen in the figures, over time it changes noticeably
and becomes more complex.
At the same time, the shape of the zone of upflows in the cloud and its incli-
nation to the OX axis change more slowly with time. As for the vertical move-
ments of air around the cloud, the figure shows that the structure of the velocity
field of vertical air currents has undergone significant changes. Firstly, in the
course of the evolution of the cloud, the area covered by the ascending and des-
cending movements of the air expanded significantly, and secondly, the velocity
field structure of these movements became noticeably more complex. As for the
maximum values of the velocity of vertical airflows in the cloud, they remain
relatively small—17.9 m/s for 20th min. and 12.8 m/s for 30th min of the cloud
evolution,
i.e.
there is a slight decrease in the value of this parameter. Figure 4
and Figure 5 show the isolines of the rate of ascending and descending air
movements in and around the cloud at the same time in the same vertical plane
passing through the center of the cloud along the axis OX. But calculations were
made for the model structure of the wind field in the atmosphere,
i.e.
for the case
when the direction of the wind velocity vector at all levels coincides with the di-
rection of the axis OX. In dark colors, the isosurfaces corresponding to the ve-
locities of the ascending currents W = 35.0 and 34.1 m/s are identified in the
figures. Comparison of Figure 4 and Figure 5 shows that the structures of the
vertical air movement fields at the points in time under consideration differ
markedly, and there is also a noticeable increase in the rate of descending air
movements. As for the structure of the field of vertical air movements, one can
Figure 4. Isolines of different values of the upflow velocity and isosurface W = 35.0 m /s
in the vertical plane (20th min).

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Figure 5. Isolines of vertical stream velocity and isosurface W = 34.1 m/s in the vertical
plane (30th min).
notice a significant change in it: the descending air currents in the considered
plane under the influence of wind in the atmosphere are localized for 30th min
in the upper part of the cloud and on its leeward side.
Comparison of Figure 2 and Figure 3 with Figure 4 and Figure 5 shows that
the velocity fields of vertical air movements around the cloud corresponding to
different structures of the wind field in the atmosphere differ qualitatively: the
structures of vertical air movements in the cloud and near-cloud space corres-
ponding to the real structure of the wind in the atmosphere, are noticeably more
complex. As for the maximum rate of ascending airflow in the cloud, attention is
drawn to the fact that the values of this parameter at the points of time corres-
ponding to the model structure of the wind field in the atmosphere are signifi-
cantly higher than its values corresponding to the actual structure of the wind
field in the atmosphere.
As the results of calculations show, at all stages of the evolution of the cloud,
the influence of the wind field structure in the atmosphere on the zone of as-
cending currents in the cloud is insignificant, which, in our opinion, is asso-
ciated with large values of the air velocity in this zone. This zone behaves as an
obstacle, which is flowed by horizontal movements of air. Based on the results of
calculations, it can be noted that the complication of the structure of the wind
field in the atmosphere, in particular, the change in wind speed and direction
with altitude is a factor hampering the evolution of convection. This may be due
to the fact that in this case the interaction of the cloud and its surrounding at-
mosphere becomes more intense: the exchange of energy and mass between
them takes a more intense character. And the flow from the surrounding at-
mosphere into a cloud of colder air with less water vapor will hamper its evolu-
tion. The values and fields of other parameters of the cloud, in particular, water
content and ice content, depend in a significant way on the structure of the wind

B. A. Ashabokov et al.
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field in the atmosphere. In this case, the water content in the cloud is localized in
the region of ascending currents, which agrees with the theory of formation of
the liquid-drop fraction in the cloud, and the level of localization of the in-
creased water content may change with time depending on the rate of ascending
currents. The region of localization of glaciation is located at higher levels in
comparison with the region of localization of droplets. It is important to note
that these areas may overlap depending on the stage of the evolution of the
cloud. As an example, Figure 6 and Figure 7 show isolines of water content at
the 20th min of the cloud evolution, corresponding to the real and model struc-
tures of the wind field in the atmosphere.
The figures show that the water-localization zones corresponding to different
wind structures in the atmosphere differ qualitatively. Significantly, the values
(including the maximum) of this parameter are also different. Comparison of
the figures shows that in the case of the model structure of the wind field in the
atmosphere, the size of the region of localization of water content in the cloud,
as well as its values, is much larger than in the cloud evaluating in the atmos-
phere with real wind field structure. In Figure 7, it can be seen that the isolines
of water content reach the surface of the earth,
i.e.
in the case of the model
structure of the wind field in the atmosphere, precipitation is observed in the
cloud. This is explained by the fact that in connection with the more intense air-
flow in the cloud in the case of the model structure of the wind field in the at-
mosphere, the formation of precipitation particles also becomes more intense.
W = 35.0 m/s against the background of the isolines of water content in the
vertical plane (20th min).
Figure 6. Isolines of water content in the vertical plane (20th min).

B. A. Ashabokov et al.
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Figure 7. The isosurface of the vertical component of the velocity. W = 35.0 m/s against
the background of the isolines of water content in the vertical plane (20th min).
The obtained results were supplemented with the results of calculations of the
maximum values of the cloud parameters at different instants corresponding to
different wind field structures in the atmosphere. These parameter values also
give useful information about the effect of the wind field structure in the at-
mosphere on the formation and evolution of clouds. Table 1 gives the maximum
values of the main cloud parameters obtained as a result of calculations. The
following designations are used in the table: Wmax, Qmax and Zmax are the maxi-
mum values of the rate of air upflows and water content in the cloud, and also
the reflectivity, HWmax, HQmax and HZm ax are the levels at which these characte-
ristics are located. It can be seen from the table that the maximum values of the
velocity of air up flows Wmax at all stages of the cloud evolution are much larger
in the case of the model structure of the wind field in the atmosphere,
i.e.
in the
case when the interaction of the cloud with the surrounding atmosphere is less
intense. Accordingly, the heights in which these characteristics are located in the
cloud are noticeably higher in this case than in the case of the actual structure of
the wind field in the atmosphere. As it is known, the importance of water con-
tent plays an important role in the formation and evolution of clouds. It can be
seen from the table that the maximum values of this parameter at all times are
much larger and located higher in the cloud evaluating in the case of the model
structure of the wind field in the atmosphere. In the same way, the maximum
glacier values behave in relation to the structure of the wind field in the atmos-
phere.
Taking into account the maximum water content and altitudes on which they
are located in the cloud, it can be noted that in the case when the direction of the
wind does not vary in height, the conditions for crystal growth and the forma-
tion of hailstones are more favorable. This can be evidenced by the values of
maximum reflectivity at different times, corresponding to different structures of
the wind field in the atmosphere. In the first case, the maximum value of this

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Table 1. The maximum values of the cloud parameters corresponding to different variants of the
wind distribution in the atmosphere.
Parameter
Actual wind field structure in the atmosphere
Model wind field structure
in the atmosphere
Time, min 20 30 40 20 30 40
Wmax, m/s 17.9 12.8 7.9 44.4 39.3 39.0
HWmax, km 3.5 5.0 5.0 6.5 7.0 6.75
Qmax, g/m3 6.63 8.7 6.9 11.8 11.1 10.9
HQmax, km 4.75 5.25 4.5 7.0 6.75 6.5
Zmax, dBZ 37.6 64.7 64.3 60.0 62.8 66.6
HZmax, km 4.25 5.25 3.75 6.5 6.0 5.5
parameter is reached at the 30th min, and in the second case, an increase in this
parameter is observed throughout the considered period of time. In addition, the
maximum reflectance value is almost always greater in the case when the direc-
tion of the wind does not change with altitude. This indicates that in the case of
the model structure of the wind field in the atmosphere, the processes of preci-
pitation formation are more intense.
4. Conclusions
A new round of research in cloud physics, connected with the study of the sys-
tem properties of clouds, is beginning.
The next stage of the development of this scientific direction should be fo-
cused on studying the influence of such structure-forming factors, the interac-
tion of processes in the clouds (emergent properties of clouds), and the interac-
tion of clouds with the surrounding atmosphere (hierarchy property). Studies on
these and other areas of the development of cloud physics and active effects on them
are possible only on the basis of numerical modeling using full three-dimensional
and specially constructed cloud models.
The results of numerical experiments have shown that the structure of the
wind field in the atmosphere, which is one of the mechanisms of interaction of
clouds with their surrounding atmosphere, has a significant influence on the
formation and evolution of clouds. These results can be used in forecasting me-
thods of hazardous weather phenomena accompanying convective processes in
the atmosphere.
Conflict of Interest
The authors declare no conflict of interest.
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