PDF Archive

Easily share your PDF documents with your contacts, on the Web and Social Networks.

Share a file Manage my documents Convert Recover Search Help Contact



43I14 IJAET0514287 v6 iss2 945to953 .pdf



Original filename: 43I14-IJAET0514287_v6_iss2_945to953.pdf
Author: "Editor IJAET" <editor@ijaet.org>

This PDF 1.5 document has been generated by Microsoft® Word 2013, and has been sent on pdf-archive.com on 13/05/2013 at 13:23, from IP address 117.211.x.x. The current document download page has been viewed 550 times.
File size: 415 KB (9 pages).
Privacy: public file




Download original PDF file









Document preview


International Journal of Advances in Engineering &amp; Technology, May 2013.
©IJAET
ISSN: 2231-1963

A STUDY ON EARTH CLOUD SPACE INFLUENCED BY
CERTAIN DYNAMICS FACTORS
Vivekanand Yadav and R. S. Yadav
Department of Electronics and communication Engineering
J K Institute for Applied Physics and Technology
University of Allahabad, Allahabad – 211002

ABSTRACT
In this paper, dynamics factors and their influences in the formation of earth’s cloud field have been studied.
These influences are mainly based on heat and water-vapour flow equations in a turbulence atmosphere. The
equations for cloud water content have been developed, considering the influence of vertical movement and
heat, cold advection and turbulence exchange. The conditions of formation and development of many form of
cloud are observed in mesospheric region.

I.

INTRODUCTION

The clouds are formed a result of the transformation of water vapour form a gaseous into a liquid or
solid state. The optical properties of clouds differ from the properties of a cloudless atmosphere. The
relationships between atmospheric (air) temperature T and clouds exceed many times the relationships
between T and greenhouse gases and admixtures, primarily carbon dioxide Matveev.et.al [1]. The
formation of cyclones (including tropical ones), tornados, strong winds, and floods is closely related
to clouds Steven M. Smith [13], Matveev.et.al [9-10], Michael A.Persinger [11] and Stubenrauch.t.al
[12].
Matveev.et.al [2–4] obtained formulas for changes in air temperature and cloud water content with
time under the action of vertical movement. In this paper, other factors have been investigated which
influence the formation and development of a cloud depending on the atmospheric temperature and
water content. The accuracy of the results of this paper is obtained by the help of MATLAB
Simulation setup.
The paper has been devided into sections: Initial Equations, Vertical Movement, Turbulence, Cloud
Formation and the Change in Water Content with Time, results and discussions, conclusion and future
work.

II.

INITIAL EQUATIONS

Before any cloud forms, water vapor must achieve a state of saturation and the relative air humidity
𝑒
must attain a value of f = 𝐸 – 100%. Since the pressure of saturated water vapor E is a function of
temperature, to estimate change f, it is necessary to use the flow (balance) equation for water vapor
and the heat flow (balance) equation.
We write the heat balance equation in the form
𝑑𝑇

𝑐𝑝 𝑑𝑡 −

𝑅𝑇 𝑑𝑝
𝑝 𝑑𝑡

𝑑𝑞𝑚
𝑑𝑡

+L

= 𝜖𝑇

(1)

Where T and P are air temperature and Pressure; R is the gas constant; 𝑐𝑝 is the heat capacity of air; L
is the heat of vapor transformation (condensation);

𝑑
𝑑𝑡

=

𝜕
𝜕𝑡

+𝑢

𝜕
𝜕𝑥

+𝑣

𝜕
𝜕𝑦

+𝑤

𝜕
𝜕𝑧

is the operator of the total (individual)derivative; u, v and w are the airspeed(wind speed) projections
along axes x, y and z(axis z is directed upward along the vertical0; t is time ; and 𝜖 𝑇 is the turbulent
flow of heat(all quantity refer to 1kg of air).

945

Vol. 6, Issue 2, pp. 945-953

International Journal of Advances in Engineering &amp; Technology, May 2013.
©IJAET
ISSN: 2231-1963
The mass fraction of saturated water vapour 𝑞𝑚 entering into the third summand in the left-hand side
𝐸
of (1) is related to the saturation pressure
by the correlation.
(𝑇)

𝑞𝑚 = 0.622E (T)/p
(2)
Logarithmically differentiating it, we obtain the balance equation for water vapor in a saturated state
(in a cloud):
By (2)
Log𝑞𝑚 = log (0.622) +log E (T) – log p
Differentiating w. r. t, t, we have
1 𝑑𝑞𝑚
1 𝑑E (T)
1 𝑑𝑝
= 0+E (T) 𝑑𝑡 − 𝑝 𝑑𝑡
𝑞
𝑑𝑡
𝑚

𝑑𝑞𝑚
1 𝑑E (T)
1 𝑑𝑝
= 𝑞𝑚 (E (T) 𝑑𝑡 − 𝑝 𝑑𝑡 ) +𝜖𝑞
𝑑𝑡
𝑑𝐸
𝐿𝐸 𝑑𝑇
=
𝑑𝑡
𝑅𝑣 𝑇 2 𝑑𝑡
461.5𝐽
Where 𝑅𝑣 = 𝑘𝑔 𝐾 is the gas constant

By (1)
𝑑𝑇 𝑅𝑇 𝑑𝑝
𝑐𝑝 =
-L [{𝑞𝑚 (
𝑝 𝑑𝑡

𝑑𝑇

𝑅𝑇 𝑑𝑝 𝐿𝑞𝑚 𝑑𝑝 𝐿𝑞𝑚 𝑑E (T)
+ 𝑝 𝑑𝑡 -E (T) 𝑑𝑡
𝑝 𝑑𝑡

=

𝑅𝑇 𝑑𝑝 𝐿 0.622E (T) 𝑑𝑝
+
𝑝 𝑑𝑡 𝑝
𝑝
𝑑𝑡

-L

𝑐𝑝 𝑑𝑡 =

𝑅𝑇 𝑑𝑝 𝐿 0.622E (T) 𝑑𝑝
+
𝑝 𝑑𝑡 𝑝
𝑝
𝑑𝑡

– 𝐿2

[𝑐𝑝 + 𝐿2

0.622E (T) 𝑑𝑇
] 𝑑𝑡
𝑝𝑅𝑣 𝑇 2

𝑐𝑝

𝑑𝑇
𝑑𝑡
𝑑𝑇

(4)
for water vapour.

1 𝑑E (T)
1 𝑑𝑝

)}
E (T) 𝑑𝑡
𝑝 𝑑𝑡

𝑑𝑡

𝑐𝑝 𝑑𝑡 =

(3)

𝑅𝑇

+ 𝜖𝑞 ] + 𝜖 𝑇 , by (3)

-𝐿𝜖𝑞 +𝜖 𝑇

0.622E (T) 𝐿𝐸 𝑑𝑇
𝑝𝐸
𝑅𝑣 𝑇 2 𝑑𝑡

= (𝑝 +𝐿

0.622E (T) 𝑑𝑇
𝑝𝑅𝑣 𝑇 2 𝑑𝑡

-𝐿𝜖𝑞 +𝜖 𝑇 by (4)

–𝐿𝜖𝑞 +𝜖 𝑇

0.622E (T) 𝑑𝑝
) 𝑑𝑡
𝑝2

–𝐿𝜖𝑞 +𝜖 𝑇

𝑐𝑝 [1 + 𝐿2

0.622E (T) 𝑑𝑇
]
𝑐𝑝 𝑝𝑅𝑣 𝑇 2
𝑑𝑡

=

1
𝑑𝑝
(RTp+L0.622E (T))
𝑝2
𝑑𝑡

𝑐𝑝 [1 + 𝐿2

0.622E (T) 𝑑𝑇
] 𝑑𝑡
𝑐𝑝 𝑝𝑅𝑣 𝑇 2

=

RTp
L0.622E (T) 𝑑𝑝
(1+ RTp ) 𝑑𝑡
𝑝2

𝑑𝑇
𝑑𝑡

L0.622E (T)

=

𝑑𝑇
𝑑𝑡

𝑅𝑇 (1+ RTp ) 𝑑𝑝
𝑐𝑝 𝑝 (1+𝐿2 0.622E (T)) 𝑑𝑡
2
𝑐𝑝 𝑝𝑅𝑣 𝑇

=𝑎

𝑅𝑇 𝑑𝑝
𝑐𝑝 𝑝 𝑑𝑡

+

𝑝

(1+𝐿2

–𝐿𝜖𝑞 +𝜖 𝑇

(𝜖𝑇 –𝐿𝜖𝑞 )
1
𝑐𝑝 (1+𝐿2 0.622E (T))
𝑐𝑝 𝑝𝑅𝑣 𝑇2

(𝜖𝑇 –𝐿𝜖𝑞 )

1

+𝑐

–𝐿𝜖𝑞 +𝜖 𝑇

(5)

0.622E (T)
)
𝑐𝑝 𝑝𝑅𝑣 𝑇2

L0.622E (T)
)
RTp
(T)
0.622E
(1+𝐿2
)
𝑐𝑝 𝑝𝑅𝑣 𝑇2

(1+

Where a =

By (3),

𝑑𝑞𝑚
𝑑𝑡
𝑑𝐸

1

= 𝑞𝑚 (E (T)

By (4), 𝐸𝑑𝑡 = 𝑅

𝐿

𝑣𝑇

2

+𝜖𝑞

𝑑𝑇
𝑑𝑡

Putting the value of

946

𝑑E (T)
1 𝑑𝑝
− 𝑝 𝑑𝑡 )
𝑑𝑡

(6)

𝑑𝐸
𝐸𝑑𝑡

in (3), we have

Vol. 6, Issue 2, pp. 945-953

International Journal of Advances in Engineering &amp; Technology, May 2013.
©IJAET
ISSN: 2231-1963
𝑑𝑞𝑚
𝑑𝑡

= 𝑞𝑚 [𝑅

𝐿

𝑑𝑞𝑚
𝑑𝑡

= 𝑞𝑚 {𝑅

𝑑𝑞𝑚
𝑑𝑡

= 𝑞𝑚 {𝑝 (𝑅

dqm
dt

= qm {

dqm
dt

= qm Lp (𝑅

dqm
dt

=

𝑣𝑇

2

𝐿

𝑣𝑇

1

𝑅𝑇 𝑑𝑝
𝑐𝑝 𝑝 𝑑𝑡

(𝑎

𝑎𝑅𝑇 𝑑𝑝
𝑐𝑝 𝑝 𝑑𝑡

2

𝑎𝑅𝑇𝐿
2
𝑣 𝑇 𝑐𝑝

(1+𝐿2

1

(𝜖𝑇 –𝐿𝜖𝑞 )

𝑝

+𝑐

(1+𝐿2

𝑝

𝑑𝑝

𝑝

𝑎𝐿2
𝐿
2
𝑣 𝑇 𝑐𝑝 𝑅𝑇



0.622E RT
𝑎𝐿2
𝐿
(
𝑝
Lp 𝑅𝑣 𝑇 2 𝑐𝑝 𝑅𝑇

dqm
dt

=

)

) − 𝑝 𝑑𝑡 ) +𝜖𝑞 ]

𝐿

1 𝑑𝑝

) − 𝑝 𝑑𝑡 }+𝜖𝑞

(𝜖𝑇 –𝐿𝜖𝑞 )
(1+𝐿2

Lp dp
)
RT dt

dp
dt

)

1 𝑑𝑝

0.622E (T)
)
𝑐𝑝 𝑝𝑅𝑣 𝑇2

0.622E (T) 𝑅 𝑇 2
) 𝑣
𝑐𝑝 𝑝𝑅𝑣 𝑇2

1

-1) 𝑑𝑡 + 𝑐

1 RT aRL Lp
(
p Lp Rv Tcp RT

RT

(𝜖𝑇 –𝐿𝜖𝑞 )

1

+𝑐

+

𝑝

dp
dt

𝑐𝑝 𝑝𝑅𝑣 𝑇2

(𝜖𝑇 –𝐿𝜖𝑞 )
(1+𝐿2

+

}+𝜖𝑞

(𝜖𝑇 –𝐿𝜖𝑞 )
1
𝐿
} +𝜖𝑞
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2

𝑞

+ 𝑐𝑚

𝐿

0.622E (T) 𝑅 𝑇 2
) 𝑣
𝑐𝑝 𝑝𝑅𝑣 𝑇2

𝐿

0.622E (T) 𝑅 𝑇 2
) 𝑣
𝑐𝑝 𝑝𝑅𝑣 𝑇2

+ 𝜖𝑞

(𝜖𝑇 –𝐿𝜖𝑞 )
0.622E 1
𝐿
𝑝
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2
2

+ 𝜖𝑞

from (2)

𝑐𝑝 𝑝𝑅𝑣 𝑇

0.622E RT 𝑎𝐿2 𝑅−Rv Tcp L dp
(
)
𝑝
Lp
R𝑅𝑣 𝑇 2 𝑐𝑝
dt

+

(𝜖𝑇 –𝐿𝜖𝑞 )
0.622E 1
𝐿
𝑝
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2
2

+ 𝜖𝑞

(7)

𝑐𝑝 𝑝𝑅𝑣 𝑇

By (6)
1+0.622𝐿𝐸/𝑅𝑇𝑃
(1-a) = 1 – 1+0.622𝐿2 𝐸/𝑅 𝑇 2 𝑐
𝑣

𝑝𝑝

0.622LE (RL−𝑐𝑝 𝑝𝑅𝑣 𝑇) p𝑅𝑣 𝑇 2 𝑐𝑝

(1 –a) = 𝑅(p𝑅

𝑣𝑇

2 𝑐 +0.622𝐿2 𝐸)
𝑝

(8)

𝑅𝑇𝑝

p𝑅𝑣 𝑇 2 𝑐𝑝
(RTp+0.622LE)
2 𝑐 + 0.622𝐿2 𝐸)
𝑇
𝑅𝑇𝑝
𝑣
𝑝

a = (p𝑅

𝑅𝑣 𝑇𝑐𝑝
(RTp+0.622LE)
2 𝑐 + 0.622𝐿2 𝐸)
𝑇
𝑅
𝑣
𝑝

a = (p𝑅

(9)

Putting the value of ‘a’ in (7), we have
(RTp+0.622LE)

dqm
dt

=

dqm
dt

=

dqm
dt

=

𝑅𝑣 𝑇𝑐𝑝

0.622E RT (p𝑅𝑣𝑇2 𝑐𝑝 + 0.622𝐿2𝐸) 𝑅
(
𝑝
Lp
R𝑅𝑣 𝑇 2 𝑐𝑝

𝐿2 𝑅−Rv Tcp L

)

𝑅𝐿−𝑅𝑣 𝑇𝑐𝑝
0.622E RT 𝐿𝑝
dp
(
)
𝑝
Lp 𝑅 (p𝑅𝑣 𝑇 2 𝑐𝑝 + 0.622𝐿2 𝐸) dt

RT 0.622EL
Lp
𝑅

𝑅𝐿−𝑅𝑣𝑇𝑐𝑝

(

(p𝑅𝑣 𝑇 2 𝑐𝑝 + 0.622𝐿2 𝐸)

dqm
dt
L0.622E (T)
)
RTp
(T)
0.622E
(1+𝐿2
)
𝑐𝑝 𝑝𝑅𝑣 𝑇2

=

RT
dp
(1-a)
Lp
dt

+

)

dp
dt

+

+

dp
dt

+

(𝜖𝑇 –𝐿𝜖𝑞 )
0.622E 1
𝐿
𝑝
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2
2

+ 𝜖𝑞

𝑐𝑝 𝑝𝑅𝑣 𝑇

(𝜖𝑇 –𝐿𝜖𝑞 )
0.622E 1
𝐿
𝑝
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2
2

+ 𝜖𝑞

𝑐𝑝 𝑝𝑅𝑣 𝑇

(𝜖𝑇 –𝐿𝜖𝑞 )
0.622E 1
𝐿
𝑝
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2
2

+ 𝜖𝑞

𝑐𝑝 𝑝𝑅𝑣 𝑇

(𝜖𝑇 –𝐿𝜖𝑞 )
0.622E 1
𝐿
𝑝
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2
2

+ 𝜖𝑞

(10)

𝑐𝑝 𝑝𝑅𝑣 𝑇

(1+

a=

947

Vol. 6, Issue 2, pp. 945-953

International Journal of Advances in Engineering &amp; Technology, May 2013.
©IJAET
ISSN: 2231-1963
a=

𝛾𝑚𝑎
𝛾𝑎

(11)

Where 𝛾𝑚𝑎 is moist-adiabatic gradient and 𝛾𝑎 is dry- adiabatic gradient. Since 𝛾𝑚𝑎 is always less
than 𝛾𝑎 , in moist-saturated (cloud) air, parameter a is always less than unity (a&lt; 1).
dp
&lt; 0 in case of a particle of air moving toward lower pressure. By (5) and (10),
dt
dT
dt



m
&lt; 0 and
&lt; 0,
dt
In saturated air, qm decreases then cloud water content 𝛿 increases
The increments of water content d𝛿and temperature dT as pressure changes are, according to (5) and
(10), related by the correlations

dqm
dt
𝑑𝑇
𝑑𝑡

=
=

RT
dp
(1-a) dt
Lp
𝑅𝑇 𝑑𝑝
𝑎 𝑐 𝑝 𝑑𝑡
𝑝

(10a)
(5a)

By eqs (10a) and (5a), we have
𝑐𝑝 (1 − 𝑎)
𝑑𝑇
aL

dqm =

𝑐 (1−𝑎)

dqm = −d𝛿, d𝛿 = − 𝑝
𝑑𝑇
𝐿
𝑎
By equation (11), it is clear that for dp &lt; 0

(12)

dT &lt; 0 and d𝛿 &gt; 0,
Also, when dqm &lt;0
Then d𝛿 &gt; 0.
Thus, the increment d𝛿 in large cyclone is significant in our study.
𝑑𝑇
𝑅𝑇 𝑑𝑝
For case of unsaturated (dry air) L = 0, then (1) becomes 𝑐𝑝 𝑑𝑡 − 𝑝 𝑑𝑡 = 𝜖 𝑇 then (6) and (11) take
the form;
𝛾
a=1; 𝛾𝑚𝑎 =1
𝑎

Then equation (10) becomes

dqm
dt

(𝜖𝑇 –𝐿𝜖𝑞 )
0.622E 1
𝐿
𝑝
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2
2

=

+ 𝜖𝑞

𝑐𝑝 𝑝𝑅𝑣 𝑇

Along the vertical, the correlation

Becomes
𝑑𝑝
𝑑𝑡
𝑑𝑝
𝑑𝑡

𝑑𝑝
𝑑𝑡

𝑑𝑝
𝑑𝑡

=

= 𝑤

𝜕𝑝
𝜕𝑧

= −𝑤g𝜌
=

𝜕𝑝
𝜕𝑡

+𝑢

𝜕𝑝
𝜕𝑥

𝜕𝑝
𝜕𝑦

+𝑣

(Static equation -

𝑝
−𝑤g 𝑅𝑇

+𝑤

𝜕𝑝
𝜕𝑧

𝜕𝑝
𝜕𝑧

(13)

= g𝜌)

𝑝

where 𝜌 = 𝑅𝑇 is the air density; g is the increase in the speed of freefall
From (5) and (13), we have
𝑅𝑇
𝑐𝑝 𝑝

= 𝑎

𝑝

(1+𝐿2

𝜕𝑇
𝜕𝑡

+ 𝑢 𝜕𝑥 + 𝑣 𝜕𝑦 + 𝑤 𝜕𝑧 =

−𝑎𝑤𝑔
𝑐𝑝

𝜕𝑇

948

𝑝

1

(𝜖𝑇 –𝐿𝜖𝑞 )

𝑑𝑇
𝑑𝑡

(−𝑤g 𝑅𝑇) + 𝑐
𝜕𝑇

𝜕𝑇

0.622E (T)
)
𝑐𝑝 𝑝𝑅𝑣 𝑇2

1

+𝑐

𝑝

(𝜖𝑇 –𝐿𝜖𝑞 )
(1+𝐿2

0.622E (T)
)
𝑐𝑝 𝑝𝑅𝑣 𝑇2

Vol. 6, Issue 2, pp. 945-953

International Journal of Advances in Engineering &amp; Technology, May 2013.
©IJAET
ISSN: 2231-1963
𝜕𝑇

𝜕𝑇

𝜕𝑇

(𝜖𝑇 –𝐿𝜖𝑞 )

𝜕𝑇
𝜕𝑡

=

−𝑎𝑤𝑔
𝑐𝑝

–( 𝑢 𝜕𝑥 + 𝑣 𝜕𝑦 + 𝑤 𝜕𝑧 ) + 𝑐

1

𝜕𝑇
𝜕𝑡

=

−𝑎𝑤𝑔
𝑐𝑝

–{ 𝑢 𝜕𝑥 + 𝑣 𝜕𝑦 + 𝑤(−𝛾)} + 𝑐

𝑝

𝜕𝑇

(1+𝐿2

𝜕𝑇

0.622E (T)
)
𝑐𝑝 𝑝𝑅𝑣 𝑇2

(𝜖𝑇 –𝐿𝜖𝑞 )

1

(1+𝐿2

𝑝

0.622E (T)
)
𝑐𝑝 𝑝𝑅𝑣 𝑇2

𝜕𝑇
( = −𝛾, 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑡𝑒𝑚𝑝𝑡. 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡)
𝜕𝑧
(𝜖𝑇 –𝐿𝜖𝑞 )
𝜕𝑇
𝑎𝑔
𝜕𝑇
𝜕𝑇
1
= 𝑤(𝛾 − 𝑐 ) - ( 𝑢 𝜕𝑥 + 𝑣 𝜕𝑦) +𝑐
2 0.622E (T)
𝜕𝑡
𝑝

𝑝

𝜕𝑇
𝜕𝑡

= 𝑤(𝛾 −

(1+𝐿

𝛾𝑚𝑎 𝑔
)
𝛾𝑎 𝑐𝑝

𝑐𝑝 𝑝𝑅𝑣 𝑇

𝜕𝑇

𝛾𝑚𝑎
)
𝛾𝑎

( a=

2)

𝜕𝑇

(𝜖𝑇 –𝐿𝜖𝑞 )

1

- ( 𝑢 𝜕𝑥 + 𝑣 𝜕𝑦) +𝑐

𝑝

(1+𝐿2

(14)

0.622E (T)
)
𝑐𝑝 𝑝𝑅𝑣 𝑇2

From (10) and (13), we have
𝜕𝑞𝑚
𝜕𝑡

+𝑢

𝜕𝑞𝑚
𝜕𝑡

+𝑢

𝜕𝑞𝑚
𝜕𝑡

= −𝑤[

III.

𝜕𝑞𝑚
𝜕𝑥

+𝑣

𝜕𝑞𝑚
𝜕𝑥

+𝑣

𝜕𝑞𝑚
𝜕𝑦

+𝑤

𝜕𝑞𝑚
𝜕𝑦

+𝑤

𝑔
(𝛾
𝐿𝛾𝑎 𝑎

= Lp (1-

𝜕𝑞𝑚
𝜕𝑧

= − (1-

− 𝛾𝑚𝑎 ) +

RT

(𝜖𝑇 –𝐿𝜖𝑞 )
𝛾𝑚𝑎
𝑝
0.622E 1
𝐿
)(−𝑤g
)
+
𝛾𝑎
𝑅𝑇
𝑝
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2

𝜕𝑞𝑚
𝜕𝑧

𝜕𝑞𝑚
]
𝜕𝑧

𝑐𝑝 𝑝𝑅𝑣 𝑇2

g
L

𝛾𝑚𝑎
)𝑤
𝛾𝑎

– (𝑢

𝜕𝑞𝑚
𝜕𝑥

+

(𝜖𝑇 –𝐿𝜖𝑞 )
0.622E 1
𝐿
𝑝
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2
2

+ 𝜖𝑞

+ 𝜖𝑞

𝑐𝑝 𝑝𝑅𝑣 𝑇

+𝑣

𝜕𝑞𝑚
)
𝜕𝑦

+

(𝜖𝑇 –𝐿𝜖𝑞 )
0.622E 1
𝐿
𝑝
𝑐𝑝 (1+𝐿2 0.622E (T)) 𝑅𝑣 𝑇 2
2

+ 𝜖𝑞 ( 15)

𝑐𝑝 𝑝𝑅𝑣 𝑇

VERTICAL MOVEMENT (CONVECTION)

From (3)

𝜕𝑞𝑚
𝜕𝑧

1

= 𝑞𝑚 (E (T)

𝜕E (T)
1 𝜕𝑝
− 𝑝 𝜕𝑧 )
𝜕𝑧

(16)

E depends only on temperature and from (4),
𝜕E
𝜕𝑧

=(

𝑑𝐸 𝜕𝑇
)
𝑑𝑇 𝜕𝑧

By Static equation (1 𝜕𝑝
𝑝 𝜕𝑧

=-

𝑔
𝑅𝑣 𝑇 2

= -𝛾
𝜕𝑝
𝜕𝑧

𝑑𝐸
𝑑𝑇

=

𝐿𝐸
𝑅𝑣 𝑇 2

𝐿𝐸
𝑅𝑣 𝑇 2

(17)

= g𝜌)
𝑝

(𝜌 = 𝑅𝑇 , R=𝑅𝑣 𝑇)

(18)

By eqs. (17) and (18), eq. (16) becomes,
𝜕𝑞𝑚
𝜕𝑧

𝐿
2
𝑣𝑇

= 𝑞𝑚 (-𝛾 𝑅

𝑔
2)
𝑣𝑇

−𝑅

(19)

By eqs. (15) and (19),
𝜕𝑞𝑚
)
𝜕𝑡 𝑤

= −𝑤[𝐿𝛾 (𝛾𝑎 − 𝛾𝑚𝑎 ) +

𝜕𝑞𝑚
)
𝜕𝑡 𝑤

= −𝑤[ 𝐿 (1 −

(
(

𝑔

𝑎

𝜕𝑞𝑚
]
𝜕𝑧

𝑔

𝛾𝑚𝑎
𝜕𝑞
) + 𝜕𝑧𝑚 ]
𝛾𝑎
𝜕𝑞
𝑔
𝛾
( 𝜕𝑡𝑚)𝑤 = −𝑤[ 𝐿 (1 − 𝛾𝑚𝑎) +
𝑎

𝑞𝑚 (-𝛾 𝑅

𝐿

𝑣𝑇

2

−𝑅

𝑔

𝑣𝑇

2

)]

(19a)

Advective heat and water-vapor flows, according to eqs. (14) and (15), are written in the form
𝜕𝑇

𝜕𝑇

𝜕𝑇

( 𝜕𝑡 )𝑎 = - ( 𝑢 𝜕𝑥 + 𝑣 𝜕𝑦),
𝜕𝑞𝑚
)
𝜕𝑡 𝑎

(

= – (𝑢

𝜕𝑞𝑚
𝜕𝑥

+𝑣

𝜕𝑞𝑚
)
𝜕𝑦

(20)
.

(21)

Logarithmically differentiating eq. (2) by x, we obtain

949

Vol. 6, Issue 2, pp. 945-953

International Journal of Advances in Engineering &amp; Technology, May 2013.
©IJAET
ISSN: 2231-1963
𝜕𝑞𝑚
𝜕𝑥
𝑑𝐸
𝑑𝑥

1 𝜕E
𝜕𝑥

= 𝑞𝑚 (E

1 𝜕𝑝

− 𝑝 𝜕𝑥)

(22)

𝑑𝐸 𝑑𝑇

= 𝑑𝑇 𝑑𝑥

By eq. (4),

𝑑𝐸
𝐿𝐸 𝑑𝑇
=
𝑑𝑡 𝑅𝑣 𝑇 2 𝑑𝑡
𝑑𝐸
𝐿𝐸 𝑑𝑇
=
𝑑𝑥 𝑅𝑣 𝑇 2 𝑑𝑥

By eq. (21),
𝜕𝑞𝑚
1 𝐿𝐸 𝑑𝑇
1 𝜕𝑝
= 𝑞𝑚 (E 𝑅 𝑇 2 𝑑𝑥 − 𝑝 𝜕𝑥)
𝜕𝑥
𝑣

𝜕𝑞𝑚
𝜕𝑥

𝐿𝑞𝑚 𝜕𝑇
2
𝑣 𝑇 𝜕𝑥

=𝑅

Similarly,

𝜕𝑞𝑚
𝜕𝑦



𝑞𝑚 𝜕𝑝
𝑝 𝜕𝑥

(23)

𝐿𝑞𝑚 𝜕𝑇
2
𝑣 𝑇 𝜕𝑦

=𝑅



𝑞𝑚 𝜕𝑝
𝑝 𝜕𝑦

(24)

By eq. (21),
𝜕𝑞
𝜕𝑞
𝜕𝑞
( 𝜕𝑡𝑚)𝑎 = – (𝑢 𝜕𝑥𝑚 + 𝑣 𝜕𝑦𝑚)
𝜕𝑞𝑚
)
𝜕𝑡 𝑎

𝐿𝑞𝑚 𝜕𝑇
2
𝑣 𝑇 𝜕𝑥
𝜕𝑞𝑚
( 𝜕𝑡 )𝑎

(

= - {u (𝑅

𝑞𝑚 𝜕𝑝
𝐿𝑞 𝜕𝑇
𝑞 𝜕𝑝
) + v (𝑅 𝑇𝑚2 𝜕𝑦 − 𝑝𝑚 𝜕𝑦)}
𝑝 𝜕𝑥
𝑣
𝐿𝑞
𝜕𝑇
𝜕𝑇
𝑞
𝜕𝑝
= - 𝑅 𝑇𝑚2(𝑢 𝜕𝑥 + 𝑣 𝜕𝑦) + 𝑝𝑚 (𝑢 𝜕𝑥
𝑣



Multiplying eq. (25) air density 𝜌,we rewrite it
𝜕𝑞
𝐿𝑞 𝜌
𝜕𝑇
𝜕𝑇
𝑞 𝜌
𝜕𝑝
𝜕𝑝
𝜌 𝑚 = - 𝑚2 (𝑢 + 𝑣 ) + 𝑚 (𝑢 + 𝑣 )
𝜕𝑡
𝜕𝑞
𝑐𝜌 𝑚
𝜕𝑡
𝜕𝛿 ∗
𝜕𝑡

=

𝑅𝑣 𝑇
𝜕𝑥
𝜕𝑦
𝑝
𝜕𝑥
𝜕𝑦
𝐿𝑞𝑚 𝜌
𝜕𝑇
𝜕𝑇
𝑞𝑚 𝜌
𝜕𝑝
𝜕𝑝
-𝑐
(𝑢 + 𝑣 ) +c
(𝑢 + 𝑣 )
𝑅𝑣 𝑇 2
𝜕𝑥
𝜕𝑦
𝑝
𝜕𝑥
𝜕𝑦

𝜕𝑝

+ 𝑣 𝜕𝑦)

𝜕𝑞𝑚
𝜕𝑡

(

=−

(25)

𝜕𝛿
𝜕𝑡

, 𝛿 ∗ = 𝑐𝛿)

𝐿𝑞𝑚
𝜕𝑇
𝜕𝑇
𝑞𝑚
𝜕𝑝
𝜕𝑝
2 (𝑢 𝜕𝑥 + 𝑣 𝜕𝑦)- c 𝑝 (𝑢 𝜕𝑥 + 𝑣 𝜕𝑦)
𝑣𝑇
𝜕𝛿 ∗
𝜕𝑇
𝜕𝑇
𝜕𝑝
𝜕𝑝
= 𝑏(𝑢 + 𝑣 )- d(𝑢 + 𝑣 )
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑥
𝜕𝑦
𝐿𝑞𝑚
𝑞𝑚
Where b = 𝑅 𝑇 2 , d = c 𝑝
𝑣

= 𝑐𝑅

(26)
(27)

Multiplier b in equation (26) coincides with the multiplier in equation (19a) if we rewrite it for the
rate of convective change in the bulk water content of the cloud, 𝛿 ∗ = 𝑐𝛿.
𝜕𝛿 ∗

𝑔

( 𝜕𝑡 )𝑤 = 𝑐𝑤[ 𝐿 (1 −

IV.

𝛾𝑚𝑎
)+
𝛾𝑎

𝑞𝑚 (-𝛾 𝑅

𝐿

𝑣𝑇

2

−𝑅

𝑔

𝑣𝑇

2

)]

(28)

TURBULENCE

According to Matveev.et.al [5], it was shown that turbulent exchange of heat and moisture plays a
significant role in cloud formation and the change in water content. Turbulent heat and water-vapor
flows, according to Matveev.et.al [6], have the form

𝜀𝑞 =

𝜀𝑇
𝜕2 𝑇
𝜕2 𝑇
𝜕
𝜕𝑇
= 𝑘𝑠 (𝜕𝑥 2 + 𝜕𝑦2 ) + 𝜕𝑧 𝑘𝑧 (𝜕𝑧 + 𝛾𝑚𝑎 )
𝑐𝑝
𝜕2 𝑞
𝜕2 𝑞
𝜕
𝜕𝑞
𝑘𝑠 ( 𝜕𝑥 2𝑚 + 𝜕𝑦2𝑚) + 𝜕𝑧 𝑘𝑧 ( 𝜕𝑧𝑚 + 𝐶)

(29)
(30)

𝑘𝑠 and 𝑘𝑧 are the coefficient of turbulence along the horizontal and vertical:
𝑐
C = 𝑝 (𝛾𝑎 − 𝛾𝑚𝑎 )
𝐿
If we are correlation (23) and expression used in [6]
𝜕𝑞

( 𝜕𝑧𝑚 + 𝐶) =

950

𝐿𝑞𝑚
𝑅𝑣 𝑇 2

𝜕𝑇

(𝜕𝑧 + 𝛾𝑚𝑎 )

(31)

Vol. 6, Issue 2, pp. 945-953

International Journal of Advances in Engineering &amp; Technology, May 2013.
©IJAET
ISSN: 2231-1963
From eqs (23), (24),(31) and (23) eq (30) takes the form,

𝐿𝑞𝑚 𝜕2 𝑇
𝑞𝑚 𝜕2 𝑝
𝑞𝑚 𝜕𝑝 2
𝐿𝑞𝑚 𝜕2 𝑇
𝑞𝑚 𝜕2 𝑝
𝑞𝑚 𝜕𝑝 2
𝜕
𝜕𝑇

+
(
)
+
+
2
2
2
2
2
2
2
2 (𝜕𝑦 ) ] +𝜕𝑧 𝑘𝑧 (𝜕𝑧 + 𝛾𝑚𝑎 )
𝑇
𝜕𝑥
𝑝
𝜕𝑥
𝑝
𝜕𝑥
𝑅
𝑇
𝜕𝑦
𝑝
𝜕𝑦
𝑝
𝑣
𝑣
𝐿𝑞𝑚
𝜕2 𝑇
𝜕2 𝑇
𝜕
𝜕𝑇
𝑘
𝜕𝑝 2
𝜕𝑝 2 𝑘 𝑞 𝜕2 𝑝
𝜕2 𝑝
𝜀𝑞 = 𝑅 𝑇 2 [𝑘𝑠 (𝜕𝑥 2 + 𝜕𝑦2 ) + 𝜕𝑧 𝑘𝑧 (𝜕𝑧 + 𝛾𝑚𝑎 )] + 𝑝2𝑠 {(𝜕𝑥 ) + (𝜕𝑦 ) }- 𝑠𝑝 𝑚{𝜕𝑥 2 + 𝜕𝑦2 }
𝑣

𝜀𝑞 = 𝑘𝑠 [ 𝑅

V.

(32)

CLOUD FORMATION AND THE CHANGE IN WATER CONTENT WITH TIME

Comparison of Eqs. (20) and (25) and (27) and (32) for the temperature and mass fraction of water
vapour shows that for all types of processes, the following equality may hold:
𝜕𝑞𝑚
𝐿𝑞 𝜕𝑇
𝑞 𝜕𝑝
= 𝑅 𝑇𝑚2 𝜕𝑥 − 𝑝𝑚 𝜕𝑡
(33)
𝜕𝑡
Since

𝜕𝑞𝑚
𝜕𝑡

𝑣

𝜕𝛿

= − 𝜕𝑡 𝑜𝑟

𝜕𝛿 ∗
𝜕𝑞
= (-c) 𝜕𝑡𝑚
𝜕𝑡


𝜕𝑞𝑚
𝜕𝑡

1 𝜕𝛿 ∗
𝜕𝑡

= −𝑐

𝜕𝛿
𝐿𝑞𝑚 𝜕𝑇 𝑞𝑚 𝜕𝑝
= (−𝑐)[

]
𝜕𝑡
𝑅𝑣 𝑇 2 𝜕𝑥
𝑝 𝜕𝑡

𝜕𝛿
𝐿𝐸 𝜕𝑇
𝑞 𝜕𝑝
= (−𝑐)[𝑅 2 𝑇 3 𝜕𝑥 − 𝑝𝑚 𝜕𝑡 ]
𝜕𝑡
𝑣

(34)

Multiplying (26) by the time increment ∆𝑡, we write Eq. (34) for the increment in the bulk cloud
𝜕𝛿 ∗

water content Δδ* =
in the form
𝜕𝑡∆𝑡
Δδ*= -b (∆𝑇𝑐𝑜𝑛 + ∆𝑇𝑎𝑑𝑣 + ∆𝑇𝑡𝑢𝑟𝑏 )
(35)
Where ∆𝑇𝑐𝑜𝑛, ∆𝑇𝑎𝑑𝑣 𝑎𝑛𝑑 ∆𝑇𝑡𝑢𝑟𝑏 are the temperature increments for time ∆𝑡 under the action of
convection and turbulence?
∆𝑇𝑐𝑜𝑛 = 𝑤(𝛾 − 𝛾𝑚𝑎 )∆𝑡
𝜕𝑇
𝜕𝑇
∆𝑇𝑎𝑑𝑣 = −(𝑢
+ 𝑣 ) ∆𝑡
𝜕𝑥
𝜕𝑦
𝜀𝑇
𝐿𝐸
∆𝑇𝑡𝑢𝑟𝑏 =
;b= 2 3
(36)
𝑐𝑝 ∆𝑡

VI.

RESULT AND DISCUSSION
When there is moisture-stable stratification (γ &lt; 𝛾𝑚𝑎 ) at fixed levels (at points), the air
temperature decreases and cloud water content increases with time with upward
movement (w &gt; 0); the temperature increases and the cloud water content decreases with
downward movement (w &lt; 0). When there is non-moisture-stable stratification (γ &gt;𝛾𝑚𝑎 ),
the signs of ∆𝑇𝑐𝑜𝑛 and Δδ*are the opposite of the signs for the case of (γ &lt;𝛾𝑚𝑎 ).
When there is heat advection, at fixed points, temperature increases with time (∆𝑇𝑎𝑑𝑣 &gt;
0) and the cloud water content decreases (Δδ* &lt; 0). With cold advection, at points with
fixed coordinates, the temperature decreases ( ∆𝑇𝑎𝑑𝑣 &lt; 0) and cloud water content
increases (Δδ* &gt; 0). On the basis of Eq. (32) we can state that during cloud formation and
development Stubenrauch et.al [12], the advective factor plays a role that is quite
comparable to the convective factor.
Under the action of turbulent heat and moisture exchange, the air temperature changes
and, as a result, the mass fraction of water vapor also changes. The third equation of (33)
is positive only upon passage to turbulence along the horizontal in a closed cold region
surrounded by a warm region.

(i)

(ii)

(iii)

VII.

𝑅𝑣 𝑇

CONCLUSIONS

The heat and water –vapor flow equations relate heat advection, vertical movement and turbulence
exchange in some extent in our region. The role of cold air is more influensive in formation of cloud
wall in a tropical cyclone owing to heat flow that the cyclone draws from the ocean, the temperature
in which (at distance of 150-250 km from the earth’s surface) increases by a few tens of degrees per
day. This growth in compensated by advection of cold air at 1950-2950 km from the cyclone. Due to

951

Vol. 6, Issue 2, pp. 945-953

International Journal of Advances in Engineering &amp; Technology, May 2013.
©IJAET
ISSN: 2231-1963
the action of this advection, powerful nimbocumulus clouds and heavy precipitation are formed in
mesospheric region.
In this paper, we focused on the fact that at the initial stage of the birth of a cloud, a significant role is
played by turbulent fluctuations in air temperature and humidity and the subsequent displacement of
volumes with different thermo hygrometric characteristics at which condensation of water vapor
occurs. At the expense of heat condensation, the temperature of the mixture increases, which leads to
the formation of vortex movement, in which colder air of the medium enters a particle of the mixture
and causes additional condensation—spontaneous cloud growth arises. This process plays a
determining role in the near-simultaneous formation of cumulus clouds over the largest part of the
skydome following clear weather.

VIII.

FUTURE WORK
(1) In our opinion, in the simultaneous formation of small cumulus clouds over the largest
part of the skydome after clear weather, turbulent mixing of volumes of air at various
values of temperature and relative humidity plays the main role. This is the future work.
(2) Fluctuations in air temperature and humidity, which increase with increasing windspeed
and turbulent exchange, are well known not only from the results of special
measurements, but also, for instance, from televised weather data. This is the future work.

ACKNOWLEDGMENT
Authors are grateful to the referee for his constructive criticism and valuable suggestions for the
improvement of this paper.

REFERENCES
[1]. L. T. Matveev and Yu. L. Matveev, Oblaka i vikhri - osnova kolebanii pogody i klimata (Ross. Gos.
Gidrometeorol. Univ., St. Petersburg, 2005) [in Russian].
[2]. L. T. Matveev and Yu. L. Matveev, Dokl. Akad. Nauk 374 (5), 688 (2000).
[3]. L. T. Matveev, Dinamicheskie faktory obrazovaniya oblakov i osadkov, Voprosy fiziki oblakov
(Gidrometeoizdat, St. Petersburg, 2004) [in Russian].
[4]. Yu. L. Matveev, Ob uravneniyakh pritoka tepla i vodyanogo para. Uchenye zapiski, no. 2 (Ross. Gos.
Gidrometeorol. Univ., St. Petersburg, 2006) [in Russian].
[5]. L. T. Matveev, Dinamika oblakov (Gidrometeoizdat, Leningrad, 1981) [in Russian].
[6]. L. T. Matveev, Osnovy obshchei meteorologii. Fizika atmosfery (Gidrometeoizdat, Leningrad, 1965)
[in Russian].
[7]. L. T. Matveev, Fizika atmosfery (Gidrometeoizdat, St. Petersburg, 2000) [in Russian].
[8]. Yu. L. Matveev and L. T. Matveev, Izv. Ross. Akad. Nauk, Fiz. Atmos. Okeana 36 (6), 760 (2000).
[9]. L. T. Matveev , Yu. L. Matveev and E.Yu.Nikolaeva “Dynamic factors in the formation of the earth’s
cloud field” Atmospheric and Oceanic optics, June 2009, Volume 22, Issue 3, pp 325-330.
[10]. L.T. Matveev, Yu. L. Matveev “Formation of precipitation” hydrological cycle – vol. II
[11]. Michael A.Persinger “The possible role of dynamics pressure from the interplanetary magnetic field on
global warming “international journal of physics science vol (1) pp-044-046, jan-2009.
[12]. Stubenrauch, C. J., A. Chedin, G. Rädel, N. A. Scott, and S. Serrar. 2006. Cloud properties and their
seasonal and diurnal variability from TOVS path-B. J. Climate 19:5531–5553.
[13]. Steven M. Smith, Seasonal variations in the correlation of mesospheric OH temperature and radiance at
midlatitudes, Journal of Geophysical Research, vol. 117, a10308, 8 pp., 2012.

AUTHORS
Vivekanand Yadav is currently pursuing D.Phil at University of Allahabad, Allahahbad211002(India) and obtained his B.E in ECE from Dr.B.R.A.University Agra, India. Obtained
M.Tech(EC) at HBTI Kanpur from UPTU, Lucknow, India. Area of interest are Filter Design,
Digital Signal Processing and Atmosphheric Dynamics.

952

Vol. 6, Issue 2, pp. 945-953

International Journal of Advances in Engineering &amp; Technology, May 2013.
©IJAET
ISSN: 2231-1963
R.S.Yadav is Presently working as Reader in the University of Allahabad,Allahabad211002,India. Obtained D.Phil from University of Allahabad,Allahabad-211002,India.Area
of interest are Digital Electronics and Atmosphheric Dynamics.

953

Vol. 6, Issue 2, pp. 945-953


Related documents


PDF Document 43i14 ijaet0514287 v6 iss2 945to953
PDF Document table of content volume 6 issue 6 jan2014
PDF Document call for papers 15
PDF Document call for papers 16
PDF Document volume 4 issue 1
PDF Document call for papers 13


Related keywords