ON THE TURBULENCE IN THE PROTOPLANETARY CLOUD BY V. S. SAFRONOV

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP80T00246A003300220005-6
• Release Decision: RIFPUB
• Original Classification: K
• Document Page Count: 7
• Document Creation Date: December 21, 2016
• Document Release Date: November 17, 2008
• Sequence Number: 5
• Content Type: REPORT

Approved For Release 2008/11/17: CIA-RDP80TOO246AO03300220005-6
ON THE TURBULENCE IN THE PROTOPLANETJ'LI Y CLOUD
By V.S.3afronov
1. The problem of turbulence in the protoplanetary cloud
is of importance for planetary cosmogony. Chaotic macroscopic
motions probably existed. in the cloud'during its formation.
Further evolution of the cloud depended to a great. extent
upon whether these original' motions damped in a short time,
or turbulence supported by some source of energy existed du-
ring planet formation. According to Kuiperts and Fessenkovts
hypotheses massive'.protoplanets formed as a result of gravi-
tational instability and turned into planets after the dissi-
pation of light' elements. Large-scale turbulent motions. with
mean velocities exceeding; the thermal velocities of atoms.
and.molecules would prevent., however, gravitational,instabi-
lity in the cloud, even if its mass was of the order of the
mass of the sun. According to Edgeworth and to Gurevitch and
Lebedinsky theplanets.grew;gradually from small condensati-
ons formed in a flattened dust disk. But even small scale
turbulent motions would prevent extreme flattening of the
disk necessary for gravitational instability0 The problem of
turbulence 'is also connected with the problem of present di-
stribution of angular momentum between the sun and planets,
as large-scale turbulence produces redistribution of matter.
and of angular momentum in the cloud.
2. The hypothesis of the preset ce of large-scale turbu-
lence in the protoplanetary-cloud was introduced by von.V7eiz
sacker [1]. But Weizsacker's arguments do not proove its
existence. Reynolds'nLunber is very large for the cloud
(about. 1010). But for a rotating medium Reynolds number can-
not be considered as-'a criterion of turbulence; Weizsaeker
regards turbulence as a result of convective instability.
But he uses the criterion of convection: for non-rotating:
medium, which is inapplicable in the case of the rotating
cloud. The problem needs therefore further study.
Approved For Release 2008/11/17: CIA-RDP80TOO246AO03300220005-6
 Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
. In order to reveal the main features of motions in a
flat, protoplanetar'y cloud- one can use the results of investi--
gationa of fluid. motion between two rotating coaxial cylin-
ders. Rayleigh [2):, Taylor [3] and Synge: [4] proved. that such
a motion of incompressible fluid is stable if the angular mo-
mentum increases outwards
This condition had to be ' satisfied .for the prot.oplanetary
t the'. pressure gradient in the cloud and
'6. own gravitation as compared with the'. gravitation of the
sun the angular momentum will be proportional to V
en condition' (1) beo.omes identical with' the condition of
stability' of` circular orbits weir known` in stellar dynamia.s'
Bt w condition (1) was obtained for an incompressible fluid
and does not take into account the- possibility of convection.
On the other hand. Weizaacker : using the criteria n of convection
left of of account the condition of stability, of circular
orbits::. These two conditions were ` a ombined in. the' paper of
the author and E.L..Rousool (5). The eondition of convection
for ,.a flat. rotating cloud (cylindrical rotation) ~:as f ou}~.c~
z d c z2) dp dA -- dP'
(w Z) a~ 2p2 d7, dz lad d z
dP d
2 -P.T. dz d
When considering small -disturbances it is possible to appxn-
ximate: smooth functions. f and., .T. for small. intervals, of
by power functions. .
1P ... wl I T ~ `? '2 (3)
Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
 Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
The condition of`, convection is then reduced to
(4)
''The prot,otlanetary cloud" being largely an -HI region one can:
take as maximum value of T in the right hand side of the ' in-
e,.quality (4) the temperature of the black body in transpa-
nt `.oloud, T. 300, where rae is the distance from the
re
e
sun in a.u:. Then
36a (5)
RTz
and the condition df convection (4) is not satisfied at any
values of al and. o,2 Hence the undisturbed protoplan.etaxy
ecl,oud is stable in respect to small disturbailoes an&.` convec--
tioan could.'. not arise ' in it at any admissible values of tem
perature' -.arid of density gradients:.
*ai' The 'possibility of large-scale turbulence during a
long time is open to 'serious objections from en:ergetia, cons --
de'rations. ' Solar"radiation entexring, the flat cloud is.. insuf-
ficient to support turbulence. Gravitations . energy of the
parts of the o loud: approa'c'hing" 'the " sun suf f ide s only, for a
short; tame:... W.eizsackere`.s value of the mean turbulent veloci -
ty of about one tenth of the orbital velocity, loads, to, the
time: of disintegration of ? the cloud.of about 14 yearn, whi-
lst for the planet formation l0 years are needed according
to WeiZsacker himself'.. It seems probable. that ' the ratio, of
the. mean turbulent,-: velocity to the orbital, and the ratio of
the mixing length to the' distance from the 'snn are of the sa-
me order of magnitude-0- Chandras ekhar: ' and ter Haar (,6) have
found e = 0.62, r from the law. of planetary. distances (but
this argument does not seem convincing) and take_ :'the'value of
turbulent velocity to be slightly 'higher., than one half of the
Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
 Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
had to damp .rap idly, According to energetia considerations on-
ly motions of a scale by :thousand times less than it follows
from Karman?s formula could exist fora long time.
nation of the planets is impossible. Large-scale turbulent: mo-
tions, if such existed at the initial stage of the evolution,
orbital velocity. Karrn gin's for:lula [71 for the nixing length
in a rotating medium leads to a still higher valu.et namely,
21= 0.8 r. Under these conditions" the time of di-
sin.t:egration of the cloud is less than 102 years and the :for-
5_ It is of interest to investigate he problem of the
the sun. Weizsacker uses shearing, stresses depending; on the
gradient of angular Vlocity
transfer of matter and angular momentum during the .existence
of turbulence in the cloud.. According to We.izsacker turbulent
friction diminished. the. angular momentum of the.rapidly'rota-
ting.inner.parts of the cloud, which therefore approached the
sun. The outer parts ao. uir.ed the momentum and went away from
dw
(6)
is unfit for large-scale.turoulent motions. Prandtl. found ano-
ther expression for the stresses as a function of the gradient
of angular momentum:
But this tensor of molecular viscositY tresses is valid
strictly sp'eaking,,. only for the case of small free. paths and
to Prandtl's and Weizsa.cker's formulae are opposite..
transfer of matter and angular momentum in the cloud according
Karman.[7] gives the same expression (7) without any comment
on Seizsacker's 'using expression '(6). In the solar system an-
gular velocity decreases with the distance from the sun while
the angular momentum.increases. Hence the direction of the
Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
 Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
Taylor (8] believes that the steady value: of' angular mom.
mentum in the central region of turbulentfhUow'(the inner`. ay-
finder' his rotating) found experimentally by and Wa,ttendo r
[91 , ''c contradicts Prandtl' s formula, as the latter ~ gives: in
this' case' zero shearing ' stresses, and would make impossible tbO
transport of angular momentum. However the equalization of amt,-\
gular Momentum-in the "main part of the flow agrees. with 3'rahd:ti
expression. The acouraoy of the experiment is .not sufficient
to state that. the derivative of angular mementum.is.exactjy
zere; We can only say that the derivative is very smaill, but
this conclusign. follows just from Prandtlts formula, if the.
turbulent viscosity is great. The same takes. ,plane ,in the 'strai-
ght flow ,in tubes. The almost flat velo6 ty prof Ile far from
the walls of 'the tube and its sharp bending near. the walls :can
be- explained, if we suppose that turbulent viscosity is high:
far from the walls, and decreases rapidly/ when.approaohing: the
walls (as the first or the -second power of the distance. from
:., walls, for example).A similar suggestion about turhulent
viscosity in a rotating flow permits. to expla:in,by using:
Prandtl' s formula:, the almost constant value of angular mq-'
mentum far from the walls and its sharp fall near the,wal
Neither the relation. (6) resulting; from the molecular* visco~-
sity tensor, nor Taylor's suggestion of vorticity conservati-
on permit to explain this peculiarity of turbulent rotational.
motion*
Probably Prandtl's formula is not quite accurate, because
of the semiempirioal character of the turbulence "t1Teory.' On
the ground of new interpretation of the mixing, length J.Wasite-
tynsky [10] has obtained the expression for stresses In ,a met
re general form. For the case of cylindrical rotation.*.,,
,L, _ 2 K CO
(e 3
Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
 Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
At Kp = 0 (purely radial exchange) one obtains
Prandtl"s formula; at (isotropy) one obtains a
formula of molecular viscous stresses with the exception that,
turbulent viscosity figurates instead of the molecular. He
found the condition of non-decreasing turbulence for inc om--
pressible ideal fluid.
f
z d (w'tt) dca
dt dz (9)
It is not. clear whether this generalization is only formal.,
or characterizes the turbulent motions more exactly.. In this
case it is not. clears .which values of the ratio }~ j~( K
are more probable in the actual. turbulent floes. For the soiaa
system the turbulence _deareases..accor-
energetic considerations. that this situation"'too'k place for
large-scale turbulence. It.might.be believed . that, small-sca
le turbulence would be. more isotropic.,But.-small-scale t.urbu-
lenee would be incompar&ble with,the theoret1ioal value df
mixing: length found. by .Barman 'fora rotating,'system, It is
not clear whether such turbulent motion4 are`possib&e,..
Being only an astronomer the author ' fails.1t.o.estimate cor-
rectly the theories.and criteria: of turbulence and should li-
ke to know the opinions about these questions of specialists
in this field. .
ding to this formula, if The si of
the stresses is then given by PrandtlV,s forddl& and.the.
transfer of matter and of angular momentum. .isopposiite to
that found by Welasacker. It seems probable according to
Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
 Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6
References
1. C.F.v.leizsacker. Zs.f.Naturforschung 3a, 524, .1948.
2. 'Rayleigh*`-, 93, 148, 1916
3. G.J.Taylor. Phil.Tran.s.A, 223, 289, 1923
4. J.L.Synge. Trans. Roy.Soc.Canada, 27, iii, 1, 1933;
Proc.Roy.Soc.London, 167, 250, 1938
5. V.S.Safronov & E.LQRouscol, Comptes Rendus de l'Aoad.
d.Sciences de 1'URSS.10#3, 413, 1956;
Problems of Cosrmo ;ony 5, 22,-1957.
6. S.Chandraseld-lar & D.ter Haar, ApJ.III, 187, 1950
7. Th.Karman, Problems of Cosmical Aerodynamics, Dayton,
Ohio, 1951.
8. G.JDTaylor, Proo.Roy.Soc.London 135, 685, 1932; 1519
494, 1935.
9. F.L.Wattendorf. Proc..Roy.Soc.London, 148, 585, 1935
J.0. J.Wasiutynski. Studies in hydrody'namnic and structure
of stars and planets, Oslo, 1946.
Approved For Release 2008/11/17: CIA-RDP80T00246AO03300220005-6