Cloud boundary determinations based upon adiabatic flo

Kibel, I. A.

English

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I 
- D E T E R M I E A T I O N  I- OF B O U N D A R I E S  
OF C L O U D I N E S S  
Dokl. A.  8. SSSR, 
Geo f ieika , 
Tom 160, No. 4, 815 -8, 
1o;datel 'stvo r r N A U E ~ r t ,  1965 
S U M H A R Y  
by I . B .  Xibelt 
The determination of boundaries of cloudiness,  as described 
here,  is based upon considering an ad iaba t i c  flow, i n  which are dissemi- 
nated separa te  repions with pseudoadiabatic processes, whgse boundaries 
a r e  not known i n  advance and must be determined alongside with the  solu- 
t i o n  of t h e  problem of motion i t s e l f .  On t h a t  bas i s ,  conditions are derived 
outs ide and in s ide  the cloud, and a p a r t i c u l a r  example is considered. 
8 
* 
I n  the  hyrdodynanic theory of cloudiness occurring from t he  leeward 
s ide  of an obstacle ,  the cloud bounGary is usua l ly  determined after t h e  
adiabatic motion near the obstacle  i s  found as the  geometric spot of poin ts ,  
at which the  moisture, t ransferred by the  ad iaba t i c  flow, begins t o  saturate 
the  spiice [l]. Pieanwhile, and provide3 no p rec ip i t a t ions  are falling, the  
motion in s ide  the  cloud must be viewed a6 pseudoadiabatic : i n s t e a d  of preser- 
va t ion  of p o t e n t i a l  temperature we must assume t h a t  of pseudopotential. 
The system of hydrodynamics equations w i l l  then be modified in its form 
and a l l  the s t r u c t u r e  of the  so lu t ion  may be disrupted. A cor rec t  statement 
J f  t h e  problem would then be the  considerat ion of an ad iaba t i c  flow, in 
which separa te  regions with a pseudoadiabatic process are  dissemniated; the  
* OB OPREDELENII GRANITS OBLACHNOSTI 
2. . 
boundaries of these regions a r e  not known in advance and m u s t  be determined 
alongside with the  so lu t ion  of the very problem of motion. 
The condition f o r  sa tura t ion  can be taken as the boundary condition \ 
a t  t h e  edge of these region. Introducing i n t o  the  consideration the speci- 
f i c  moisture q ,  we shall write a t  boundary q = q,, (T, p),  where q,, 
is determined by the w e l l  known Idagnus formula and is dependent on both, 
the temperature T and pressure p 121 : 
(1) c,,,(T)=G,1.10-3P csp 17,13 - Q,:m Y r-35- 
i? T - 273 = 0,23.2J-S- emar(T); 
We s h a l l  consider f o r  the sake of s impl i c i ty  the case of s t a t i o n a r y  
flow pas t  the c res t .  The motion takes place in the  p l a e  ( x , e )  
the hor izonta l  and t the  v e r t i c a l  coordinate 1. Fron the co r re l a t ions  
( x being 
(2) 
we may introduce the current  f w c t i o n  9 .  
\de disregLard tlie e f f e c t  of air compressibi l i ty  and we l i m i t  the 
notion by the s t r e a s l i n e d  contour from below, and by the  hor ieonta l  wall 
t 3 H from above. L e t  then he given for t h a t  value u, of the  ho r i eon ta l  
v e l c c i t y  u a t  
n 
sing gressure from them, will Five the co r re l a t ion  [33 
x = - 00 and the value T, of the temperature Tr urn 3c U ( t ) ,  
=To - y z ( t ,  T 0 are constants).  The equations of motion, after exclu- 
where 51 = ' ' I  / dz - dw / ax = A$, T' is the def lec t ion  of temperature from T ,,; 
g is tilo yr: --it<: t i o n  accelerzt ion;  T, is the average temperature of the  
z i r  (Tn* 250'). The equation of heat  inflow w i l l  be wr i t t en  in t he  fo ra  : 
?.a = : ( z  - 1) /z ]R/f l  ( B  is a ~ ~ 1 s  constant, 3~ i s  the  heat  capaci ty  
r 2 t i o ) ;  ~t tkc 7 ;  t i n e ,  vie have outside the cloul! E = 1 ( see [2]) and 
(5) 
i n s i d e  the cloud ( L  b e i n r  t h e  l a t e n t  condensation heat ar?d c - the  hea t  
caDacitv of t h e  a i r  a t  c o n s t m t  D r e s x r e ) .  
? 
t 
3. 
For the  determination of moisture q, w e  shall take the  t r a n s f e r  
e quat ion 
which is in tegrable  and gives 
of g, 
n i t y  ahead of the  obstacle.  
Q = Q (9, where Q is an a r b i t r a r y  funct ion 
of which the form is found from t h e  condition q = q, ( t )  at i n f i -  
Introducing Q (q) i n t o  the left-hand p a r t  of (11, we s h a l l  ob ta in  
a t  cloud boundary an interconnection between $ p and T. Then we may 
s u b s t i t u t e  with a great degree of prec is ion  p i n  (1)  by its value 
n 
z 
where po is the  pressure at sea l e v e l  , and write ins tead  of (1) : 
correspondingly t o  4, e, T' at cloud boundary). The equation 
(+~tp, zI;p, T'icp 
( 7 )  interconnects  c ] , ~ .  vim. Zowever, the temperature T'  may be excluded 
with t h e  he lp  of (4).  Ls is well known, for the  region outs ide  the  cloud 
(4) adnits the  i n t e g r a l  
Y )  [. - f :  ($; I ,  (8) ri = - iY3 - / 
i!i 
_------- 
3(---J where f l  is an a r b i t r a r y  funct ion of V, 
L '. the  form of which s h a l l  be found from the  
condi t ion a t  i n f i n i t y .  On t he  o ther  hand, 
w e  may subs t i t u t e ,  with t h e  same degree 
v '. - 
IqS 
- 
I of prec is ion  with which (5) w a s  wri t ten ,  
V 2 = - 7  1 
/ / ! 
1 
in t he  function p and T by t h e i r  
values p oo and T, *. 
1 5 -  10 4s 
Fir .  1 
l? 
RTm * Prac t i ca l ly ,  s u b s t i t u t i n g  poo by the approximate formula P- 
we may obta in  f o r  C the apvroximate expression 
(--- 
where 
E a E- = (1 + 0,133 CZP X) (1 f 0,851 crp %)-I,  
4. 
Ve m q  then write f o r  the  cloud 
(9) 
where F1  is st i l l  another a r b i t r a r y  funct ion of V.  
Introducing T' from (8) i n t o  (3): we shall nbfs in  entelds ef the 
cloud a single equation for admitt ing an i n t e g r a l  of the form 
where f2 ( \V I  is a new a r b i t r a r y  funct ion of 4'. 
Ins ide  the  cloud 
(11) 
where F (V) is an a r b i t r a r y  function of V. 
2 
In the following we s h a l l  u t i l i z e  the condition (7) in the approxi- 
mate form* 
-> 
Q(I&)= .3.s.:G-"-cs;,(Lc,f I ' U  bT,;;-ca,,,), (12) 
where . %,2 = by - -'- = - - t,, -- T, - 273. 
J{ I  I , ,  ' I , ,  ' 
q p  in the form If we represent  q Qo 3 qo exp (- f't) ,  where qo 
and r are constaxts,  we may write, according t o  (81, qm=q0exp(-rft(goo)). 
since T' 
Q(9) =-qoexp(-rfi(q)). Zubst i tu t ing  t h i s  Q into (121, and T' according t o  (81, 
we .sfla11 have t h e  f i n a l  coni i t ion  , linking ~ K P  a d  QKp in the form 
becoaos zero far off the  obs tac le  and we have [fi($)lz,,=z.Then, 
ZiiP = rnf, + s, 
where 
Subsequeztls, d i r ec t ing  x t o  -00, we s h a l l  have, according t o  (10) 
We s h a l l  requi re  a t  cloud bouncary t h e  cont inui ty  of funct ions '  
n These t ransfer . ,  and the same goes for t h e  ve loc i ty  and the vortex 
binding condi t ions w i l l  give us, first of al l ,  t he  p o s s i b i l i t y  of l i n k i n g  
- 
* We approximately s u b s t i t u t e  the i n t e  ation and we reject  T' in the 
denominator OP the  right-hand pa r t  of 'F 7 ) .  
a 5. 
F1 and F2 with fl. equating T' by (8) and (9) at (131, we s h a l l  obtain: 
Therefore, F1, F2 and f2 are expressed through fl. A6 t o  the defer- 
n ina t ion  of the la t ter ,  and according t o  (21, the equation 
i+) 
1 
Introducing a t  the  same time dimensionless quan t i t i e s ,  we shall 
(i)= \ U ( z ) d z .  (18) 
se rves  i ts  purpose. 
f i n a l l y  have outs ide the  cloud 
and in s ide  the cloud 
where 
(V being the  c h a r a c t e r i s t i c  velocity).  
A 8  an example, we s h a l l  consider t h e  ca8e of a very s l a n t i n g  ob= 
stacle,  when the  longwave method can be appl ied ,  and in the  equations (1) 
t h e  second der iva t ives  by If can be dropped. Le t  u8 assume also t h a t  
U ( t )  = const = V. Then f (\y) = \y. W e  s h a l l  take for 2 a constant m e a n  
value. I n  the  absence of cloudiness we s h a l l  simply obtain 
!I' = Z - Zo csc D( 1 - 20) sinD(1-Z) (22) 
(Z = 2, (X) being the equation of the streamlined contour). 
But, i f  a cloud should appear above the  obstacle ,  we would have 
to estimate : under the  cloud 
Here Z1 and Z2 are  rzspec t ive lg  tLe lower and upFer limits of t h e  cloud 
b = v l  - m%D. The qnmtities Z1 and Z2 are determined from a system 
of two t ranscendent ia l  equations (we s h a l l  omit them), which is obtained 
provided : a) we equate 
and b) we equate d Y / d Z  from (24) and ( 2 5 )  at 53 3: 22. 
whether or  not  we shall have n >'i (e.\la < yj  and n < 1 (EYa > y ) .  In Fig. 1 we 
gave an example of cloudiness contour a t  flow past t he  obstacle  2, = 0.42eox; 
i t  was assumed t h a t  D = 3, m = 0.6, S = 432. The cloudiness contour for an 
a l l  adiabatic flow is given by dashed curve; all the  o ther  l i n e s  represent  
t he  cloudiness at n = 0.9 and n = 1.5, respectively.  
d Y  1 aZ from (23) and a V / I Z  from (24) at 2 = X1 
The so lu t ion  will have a sharply different character  depending upon 
World Meteorological Center Received on 28 October 1964 
CONTRA@P NO. NAS -5-37 60 
Consul tants  & Designers, Inc. 
Arlington, Virginia 
Translated by ANDRE L. ERICHAIJT 
on 27 August 1965 
R E F E R E N C E S  
[1].- A, A. D0RODNITZYN.- Tr.Glavn.Geofiz.Obs. V. 23, 1938. 
[2 3. -  I. A . KIBELt.- Vvedeniye v gidrodinamicheskiye metody kratkosrochnogo 
prognota pogody (Introduction to the hydrodynamics' 
3 ntr, metnocis of weatbet f~recmtiizg). E., r 7 / f O  
133. - N. E KOCHIN, I. A. KIBEL' , No V. ROZE. - Teoreticherkaya g5.drodinamik.a. 
(Theoretical Hydrodynamics).- ch.1, M . 1963. 
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Determination of boundaries of cloudiness

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