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28 July I960

THE ROLE OP TURBULENCE IN THE TRANSPORT

OF AN ATMOSPHERIC EDDY

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1

CSOt 1»269-N

THE SOLE OF TORBÖLENCÄ IN THE TRANSPORT OF AN ATMOSPHERIC EDDY.

/"«Phia is • translation of ah article written by L.üa. Byshakov 4 in Problem* Arktik! % Antarktiki (Problems of tha Arctic and

Antarctic), No. 1, J*ningrad, 1959t/

The idea of conceiving atmospheric movements as eddies of varying o

dimensions appertains to the eminent Ritssianmeteo^ogist P.I, Brounov

and was formulated by him at the end of last century (ß/. Nowadays this

concept has found universal recognition, inasmuch as it establishes the

likeness of such atmospheric phenomena as planetary eddies, active cen-

ters of the atmosphere, moving cyclones and anticyclones, local circula-

tion, etc.

An important problem of modem meteorology is the problem of the ge-

nesis, development and motion of cyclones and anticyclones which consti-

tute one of the fundamental links of the general atmospheric circulation.

Enormous masses of air take part, in the motion within the cyclone-anticyc-

lone system. The life cycle of theses eddies is rather complex. The pre-'

determination'of the variation in intensity of cyclones and anticyclones,

their trajectories and the velocity of vertical propagation of the distur-

bances is very difficult because it depends on many factors.

Present-day research on the dynamics of large-seäle atmospheric pro-

cesses is based on the eddy-transfer equation (Fridman's equation). Upon

analyzing this equation, l.T. MatveyeY /7/ formulated rules for a qualita- *

— 1 ■ —

'S....:.-,;

sre evaluation of the'conditions for the origination, intensification

d motion of cyclones and anticyclones. Let us write the eddy-transfer

nation in the follovdng form A* ch,i?/s

! - <^+*.>(£+$+H(£-t)- \ a) Here, Ü and V are the components of the wind velocity on the axes x

& y, respectively (the X-axis points eastward* and the y-axis north-

Td)iQ is the vertical 'component-of the relative vorticityj Äis the

■riation with latitude of Coriolis» parameter; 0 is the air density;

id$T , T■ ai*e the components of the shearing stress of the turbulent * . y ■

irces«

It ensues from Eq. (1) that the variation of the vertical component

: the vorticity in a certain region is determined by the following fac-

>rs:

A. The horizontal transport (advection) of an eddy from adjacent re-

.ons (1st addend). This addend is positive in the case of predominantly

rclonic vorticity of the air stream transported; it is negative if the

>rticity is predominantly anticyclonic.

B. The geostrophic advectiamof heat or cold (2nd addend). In the

se of the foimer, horizontal baroclinicity contributes to the formatin

d subsequent intensification of a negative (anticyclonic) eddy, and

"wrnm^

W' Q^

in the case of the latter - c? a positive f,rfv,,,^ ,. ' -'"'••-„ #

CU i&iddional eddy Kolion- (3r0 addend), ?ht. ... . n' -°t;<,02i of an eddy ha-

ving an equatorially directed ccwiporent produce an -,<■ , ■ •-e<-- ivhich tends'

to intensify a cyclonic eddy txid ic uvaken a; anticyclone «,. *.

warcdly directed meridional component- provokes hh^ opposite efU-.t.

D. Horizontal divergence of th« vind velocity (.'Ah add%nd).. A po-

sitive divergence tends to intensify an wishing .anticyclonic eidy, where-

as a negative-divergence (con-rergenct) tends to integrity ^ existing eye»

lonie eddy,

E„- Turbulent friction (5th addend, which will he dis?.u?.«ea :x,-< more

detail below). '.

In ref„ $£, L,T. Matveyey and v.*., ihyabrikov, using, factual data re-

lating to 46 cases of eddy formation, calculated the follovdng quantities:

1} the individual variation of the eddy in geostrophic approximation from

the FoifTiula ■

at ) 2u.f ' ut~ * • s

'.vhere .£,F is the pressure.Laplacianj -2} the geostrophic temperature ad-

vectionj 3) a factor taking account of the latitude effect. The caldila-

tions vere effected at sea level and pressures of 850, 700 and 500 mb.

Analysis of the mean values given in the reference work shows that the

above quantities .diminish vdth altitude; notably, at 500 rib each of them

decreases on the average by one-half as compared to the ground value.

The baroelinic tera to £q. (l) has the esae order;of maynlhud" as the

iivdividual variation of the eddy with t:hne and the addend accounting for

the altitude effect. As regards the divergent, addend, the authors/S./

state a number of reasons in support of the view that its effect in the

process of eddy formation is inconsiderable» Unfbrbunat-ly, ':■:■■: ::.: -;-Oö

possible to calculate this addend with a sufficient decree of accuracy«

Thus, le. P. Borisov/'l/ showed that when calculating; the -.-and velocity

divergence on barometric data from topographic maps, then, at different-

ia directed axes of the coordinates and various values of the different!-

ation step,the results obtained may differ twice as Ruch for one. and the

same point.

By calculating the addends of 3q. (i) the author of the present psper

inveilirated the process of intensification and propagation of an anticye-

. lone which on. February 1st 1956 was located ovea^ Scandinavia and had a pres-

sure of 1047 nib- in the center. During the period fror 1-4 February this

anticyclone shifted to the issue of the river Ob. The pressure in its cen-

ter rose to 1058 mb. While the anticyclone was moving eastward, the pres-

sure tendency at individual points of its trajectory reached 22 mh per day.

The region conprisinr Surope and West Siberia north of 47° of latitude as

well as the North Sea, . Norwegian Sea, Barents Sea and Kara Sea wae ecvo-

rpd v/ith a network of points, • for each of which we calculated the values

for the vorticity in a gecstrophic approximation and the three addends

from Sq. (1) determininpjthe temperature advection and the latitude effect.

./the adjective transfer of the ?-ddyj ^

The calculations were performed for each day and for all levels down to

a pressure of 20Ö mba' Comparison of the -allies for the addends calcula-

ted -at- the different levels confirmed the view that, together with the

eniperature advection, the baroclinic term plays a deteralnative part in

the local variation of the eddy. It appeared that sfc a level relating

to isobarie surfaces of 300 and 200 mb the values of the addends accoun-

ting for tern parature advection frequently exceed the correspondinC va-

lues in'the ruddle troposphere. Comparison of the aggregate values for

the three addends at various levels with the ground valuer of the diurnal

pressure tendency justified the conclusion that under certain conditions

the variation' of ground-level pressure if! determined basically by the pro-

cesses of vortical and tern perature advection (allowance made for the la-

titude effect) in the upper troposphere and in the lower stratoppherfi.

Such a conclusion signifies tint there arc two layers in the tropo-

arhore in \hich the processes of e-'dy fownation develop most actively,^.,

the lovrer troposphere and the layer that encloses currents. It is natu-

ral to a;suwe that vertical turbulence generated' in the active lay^r is

-.und to involve the adjacent layers in the rr.otiori. Indeed, the trans-

port in the atmosphere of any substance as vieles of monventua proceeds,

;r^rl-, bv turbulent interchange. It is taiovn that in the bottom layer

of the atxnosphere the shearing stress of. the turbulent, fractional forces

decreases with altitude. The.action of'theses forces also operates to

weaken a cyclonic or anticyclonic eddy. However, in the free atnos-

phere, the forces provoked by turbulence do not disappear, bit rather

5

P. I. Brounoa /?./ held that it was necessary to u^tady not only

the translationai motion of atmospheric eddies vdthir the hystsa of

currents of the aenaral circulation of the atmospherej hut also trap

conditions of transfer of the vortical, disturbances fmr.i laysr to lay-

er«

Academician K.Ye. liochJn /5, .67 first examined the affect of turbu- .

Isrit friotional forces on atmospheric raoveaenis of vide scope, In design-

inc, a model of tiro saneral atmospheric circulation, 11,Ye. Köchin concei-

ved the vrhole troposphere as a boundary lever and tine individual., cyclones

and anticyclones as turbulent eddies« It was demonstrated- thai, fractional

forces materially affect the nature.,of air currents not only in the bottom

layer,' but also in the free atmosphere. M. Ye« Köchin proved that a theo-

v- ->o3tulatino a stationary zonal circulation uith meridional end vertical

,~r^.0i0o^v ^omao-ents ma-y orfla be evolved orovided the forces of turbu-

lant friction ^re allowed for.'

Lat us transform the 5tb addend' in So. (l)

usinr the-Expressions fduid for the components of turbulent tan-on+ial

- -hJ1'

and

*>~*1»' ; (3)

where k is the coefficient of turbulence. Neglecting the variations

. in density and in the coefficient of turbulence in the horizontal plane,

we obtain the following Expressions for the components of the shearing,

stress of the eddy:

■>,■,»" dy "rdy*x *

S£"**!** -(u) '-M

Let us substitute Expression.' (k) into Formula (2)

(*).-f[i(*»Ä)-i(*'-Ä)]- (5)

Introducing the vorticity, we get

(-3r)x = T-*(k>-^)

The turbulent stream of the vorticity in the direction,

determined by the Expression (ky ch« * )

is

,^r—,*#. (75

vhere fl1«1' is the averaged product of the eddy fluctuations and

the vertical velocity«

Consequently, according to Sq. (6), the formation of eddies under the

influence of frictional forces is determined by the derivative vdth res-

pect to height from the vertical turbulent stream of the eddy, . For a qua-

litative analysis of the conditions of. variation of the eddy, let us dif-

ferentiate over % in Eo, (6)$ we then obtain

(d)

It ensues that the velocity of transport of the eddy in the vertical

direction is determined by the peculiar distribution along the vertical

Ci- • Thus,■ at the origination of a high-altitude cyclone the perturba- z

tion is transmitted downward more rapidly in those layers where the gra-

dient CL is larger in terms of absolute magnittide. and where this gradient

falls,off raore abruptly from one level to another. On the other hand,

the fomation of an anticyclonic eddy at -high altitude is conducive in due

course to -the origination or intensification of anticyclonic circulation course w> 0 vertical gradientjl z is larger and rises; in the next lower layer in a spot whereforeT qülcKiy m Ü duWIiUaf'tf di-

In addition,? tbe velocity of 'vertical, 'transport of -the odd.-1"!;:;'

*~. *

hr

;^:i. %'■ u)<-- decree of- tiirUürRce cf th^ st^o^'pliet^«, The pe-rtvu-ba«-

t1i>r 3 s transmitted - to the nBiidd-oririp: layers at a syendis"? rate ri^sa the

r.,•.■,-,'•■ y?5f^t /-.fr tuvbulons^ arid its 'V«rttcal yradieirt are Xarya*

■JV, o'"^'-i'i'> tr< iViustratn tho effect o.f turbatleBtyfrirtioudi. .üoro?r3 on

v-"i-' v^rtie,-:! propr-dtdon of an ed<3.j> -we plotted contoi^c ?arv--<c traeiny •

)-;,,-. ,r.V.->,,, f,f- yhc- '<;;•;• i.tcit"1" and ooofficiant. of turbulence for the os.^o of thO •

.arit.'"*c"-"f;iriTie on "? febrnarv lc56 ir its rypirn of bc'vyy;. Zwi«

-.->-. a .-.-f ■-• r>----:'-r\ nnf en 1/j-ld Kovetiber 195? CV;?T the Barents Se-'i (Fig*l)

vr^bd.-.dt^ "vafi eateulaiA>i dnygeuctropbie apvrosnjaaticr^ v;Mlo '-he co~

'.?••. bo rer:ontlv, t>.f!r--;- ordrrted DO meM'od for cater let inp 'the coe-afi«*

cd a>a.av -:t^ prbliaaba'.r of L.V. K-iKv-yov'o vrortc /?,, 10/, 5a vided

*«-f |2.M4IKP-1.15? »gfo'- ?> - 08O72j,

■ftWf . ^^Y

adiant nf !-;■-• vdad. velocity in m/f.;ect tonj

f ^r/Ss xz --iho vertical tanperature gradient in °C/lanj

y io J-.he drp adiabatio temperature gradient]

c '.c the nean ;:ind velocity in the layer.for vdiieh k io toil ; es,, p.r*«-

^r,-^"l^ foV'pvrvre^e^c the turbulence of the free atniosohere ac a P

ctiori of its theraiodynamic stats. This function vas found in the

rss of processing the factual data on overloads for an airplane in .

3 horizontal- flight and fro;;! data on temperature and rind 30rmdir.es.

flights a-ere undertaken with airplanes equipped vdth accelerographs

altitudes of- 1 - 2 and up to 1" km. This provides a basis for ueing

aula (9) for calculations of the coefficients of turbulence throngh-

•the stratosphere and thelovrer troposphere. . .

-•• i

1. Development with height of the vertical component. o£ the vorti- i+#,r i* tbe genesis of a) an anticyclone and In)a C/CJ-G^

1 0

Table 1 gives the valuesof the coefficients of turbulence calculated

■on barometric data from topographic maps Tor layers situated better le-

vels of 850, 700, 500, 300 and 200 nib. The firct of the four casro con-

sidered, relates to the period of active anticyclone .genesis in Februa-

->-■ io^ r^bui ated are the values of the coefficient of turbulence over

the central region of the anticyclone on 3 Febraary.

Coefficients of tuu%ulei^e_j^^

.3*

i CIOII aT«octJ>epu ,

(mo) ' -j . a

73Hc.iu.,4üÄB.x <58cni..* "•*ilü >-. ; , — „a 4——. , i — i I i . i _• , I, *

.«•»I'M« * i t M M * * !— | ■ i '

4.0 8.0- 4.4( 4.0j 7,5 11.o'».6 6.6 3.5 700-500 ! 6,0

500-300 I 7.0 300-200 ! -

5.0 -'.5

13.« 2.4 2I.0' 5,7 23.0! 2.0 16,0,30.7, 6.0 0.9

_| _ ' 0.41 1.8123.0J 2.0

6.0 7.0 4.0

6.3 65 2,5

19.0 61.0 3.0 22,0 94.4 8,0 28.0111^4.0

O.0I 3.4130,0 »8.0;°.°

% ! c * I

7.6 j 20.01 495 4.2|2S.ojlöü.O

2.3129,0! 4.0

1-Atmospheric layers (jnb)j 2-N. lat.; 5-E. long. 2l Uctcj vis pdven in O/km, j& in a/sec* to, c. in Vsec, K xn m /eec,

TiKU 2 presents the values of the factors cS eddy formation at various

levels on the same cay. Whereas conditions in the lower part of the,

troposphere ( data at -round level and at £50 pb) are apt to intensify an

anticyclonic eddy, the temperature advection factor at level? of 300

and. 200 nib indicates favorable conditions for its abatement. However,

on k February the intensity of tne anuxc0'uioa.,.c w-,, ,-<=,-— „

in the upper troposphere aid in the lower stratosphere, but aLi-o at

levels of 500 and ?00 mb (FiC.l). Both at 850 nb and at ground level the

antic-clonic eddy intensified. The answer to the question as to why the

1 1

abatement of the anticyclonic eddy spread to the middle troposphere,bub

failed to reach the- ground layers is to be fo^md in the distribution with

height of the coefficient of turbulence on 3 February, In effect, it

appears that strong turbulent currents had developed in the atmosphere

at levels upward.of 700 rib. Under these conditions the influence of fric-

tions! forces on the process of eddy transfer proves to be substantial.

It is imagined that the weakening of the anticyclonic eddy in the middle

troposphere may be explained as a result of the perturbation originated

at the higher levels. The layer of 850 - 700 rib, by contrast,^ .may be

described as an "arresting" layer,' since here'the moderate growth of

turbulent interchange shielded the lower troposphere from the influence

of the bisher layers. Table 2

Factors of eddy formation according to barometric data from topographic maps

Factors of eddy formation

•

3/11 1956 r. 73* cm., 40° B.X

14 XI1957 r. 75" c. a., 30» a.x

15/X1 1557 r. 75s c. m., 30* B.A.'

16/XI1967 r. 68* c.*t, so* s.a.

i T \ • <* ' as!

i 1

. ■ t

'

. I 1 x to» sec"2 •

At sea level . 850 jab "':'"■■

TOO . Ö03 .

. 300 . 200 .

1.2 . 0.7

1.« -2.1 j -2.6 -M

—1.7 -33 -2.1

0£ U 0.7

-«.6 -63 -1.1

22 1 1*

1.4 i

-3U5 -5.4 -0.4

3* 2,4 0,2

-SV\. -10»cf<"2 4»

Ha vpome »op« SS0'.jab 700 .

0.6 003 0.01

■ —

64 3.4

1.7 *

Ifi 0.4 0,2

. 500 . — — — <H2

300 . — — i __ —

200 . — . — — —

12

Tables 1 and 2 also furnish data relating to the process of cyclone

genesis observed over the Barents Sea on 14-16 November 1957» On 14 No-

vember observations revealed a weak pressure field at ground level in the

region of the Barents Sea. As could be concluded from the barographic

maps, this region lay to the left of the axis of an intensive current

(with a maximum velocity up to 220 lon/hr at 500W) proceeding from the

northwest in the region of the Greenland'Sea toward the region of the

middle Volga in the southeast. As seen,from the map, on 15 November at

3 a.m. a strong, widely extended cyclone had developed over the.Barents

Sea, The pressure in the center of the cyclone, at sea level, amounted

to 982 nib, as against 1013 mb measured at this point 24 hours previously.

An intensive cyclonic formation was observed over the ground center at a

.Ifivel of 350 mb, but later on this perturbation faded rapidly with height.

Twenty-four hours later, on 16 November, the ground center of the cyclone

had moved 1100 \<m to the southeast. Pressure in the center of the cyclone

fell off further by 10 mb. The magnitude of the positive eddy increased

appreciably $& a level of 850 mb (Fig.l). The cyclonic reconstruction of

the pressure field at levels of 700 and 500 mb.proceeded at perceptibly

decelerated rates« Above the 500 mb level an intensification of the anti-

■ cyclonic eddy was noted. Thus, despite the origination of a powerful

cyclonic eddy in the lower troposphere, its transfer to the higher lay-

ers was rather limited. This phenomenon can be.explained by way of ana-

lysis of the conditions of turbulence in the atmosphere. It appears that

daring all the days under examination the coefficient of turbulence was

- /3 -

* high in the lower and middle troposphere, but was low in the upper

troposphere .and lower stratosphere.

The examples discussed above, show that under certain conditions the

fr.ictional^ forces in the free atmosphere may noticeably affect the trans-

port alike of positive and negative eddying currents in the vertical di-

rection.

The calculated characteristics of eödies and of the coefficient of

turbulence make it possible to appraise the order of magnitude of the A -* ' the vortical flow £o3^.various

term (d£l /tft)_, from Eq, (6). To do this, we. first evaiuatedVfrom

&q. (?) and thereafter the variation of this flow with height. It was

found that the term (&Cl,/vt ) has an order of magnitude equal to 10 --.

and, hence, is comparable to the values of the baroclinic and la-

titude terms«

References

1. Ye« P. Eorisenko. "On the Horizontal Velocity Divergence in the Atmos-

phere." Meteorology and Hydrology, No. 6,1950.

2« F. -I. Brounoy», "The Movement of Cyclones and Anticyclones in Relation

to the General Atmospheric Circulation, and Weather Forecasting«'5 Me-

teorological Bulletin, 1Ö92.

3. F. I. Brounov. "On the Theory of Thunderstorm Eddies». l"b., 1897.

4. I.« St Gandin, D. L. Lykhtxaan, L.I. Matveyev, M.I. Yudin. "Fundamentals

of Dynamic Meteorology." Gidrometeoizdat (Hydrometeorological Press),

Leningrad, 1956«

5« N.Ye. Köchin.*0n the Simplification of Hydromechanics!.Equations for

14

the Case of the General Atmospheric Circulation» 'Works of the Main

Geophysical Observatory, Bull. 4* 1936»

6, N. Ye» Köchin« "Construction of a Model of the Zonal Atmospheric Cir-

culation»*. I~b., Bull. 10, 1936.

7« L. T. Hatve.yey. "Rules for a Qualitative Analysis of the Conditions of

Eddy Formation in the Atmosphere and Some Results of their Verification",

Meteorology and Hydrology, No. 7> 1953.

8. L, T, Matveyev, V.A. Zyabrikov. "On the Qualitative Analysis of the Con-

ditions of Eddy Formation in the Atmosphere."j/No. 7» 1958.

9. L. T. Matveyevj "Investigation of the Turbulent Structure of an Air Flovr

in the Region of Lake Sevan by Means of an Airplane." Works of the Main

Geophysical Observatory, Bull. 78* 1958.

10. L.T. Matveyev. "Quantitative Characteristics of Turbulent Interchange

in the Upper Troposphere and Lower Stratosphere." Reports of the

Acad. of Sc, Series on Geophysics, No. 7, 1958.

Received 4/l5 1958

END

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