propagation of mhd shock waves into regions of various uniform densities

10
IL NUOVO CIMENTO VOL. 27 B, N. 1 11 ~aggio 1975 Propagation of MHD Shock Waves into Regions of Various Uniform Densities. S. CHA~DRA and K. M. SRIVASTAVA (*) Department o/ Mathematics, University o/ Jodhpur - Jodhpur (ricevuto il 28 Ottobre 1974) Summary. -- In the presence of uniform magnetic field in an infinitely conducting fluid, the pressure behind the MHD shock wave is investigated after the shock wave has interacted with the density variation parallel to the shock front. The case is generalized to another density variation perpendicular to the shock front. The density variation is due to a small discontinuity jump across the interfaces. The problem is formulated under the assumption of small change in density and the shock profile is obtained. It is observed that the curvature of the shock increases as the Mach number decreases or the strength of the magnetic field increases. 1. - Introduction. The interest in plasma shock waves was stimulated by the possibility of heating ions rapidly to thermonuclear temperatures. Lately, the interest in shock waves has centred on geophysical and astrophysical phenomena. In the laboratory, the pinch effect is generally used to compress an initial plasma, thereby generating a cylindrically imploding shock. Iqoneylindrical pinches occur in very short pinch tubes (1), toroidal pinches (2), plasma guns (8), and plasma focussing devices (4). Simulation of geophysical shocks has also been per- formed in a plasma wind tunnel driven by an are source (% Satellite observa- (*) Present address: Institut fiir Plasmaphysik der K.F.A., Jiilich. (1) A. E. ROBSON and J. Sn]~FFI~LD: I.A.E.A. Novisibirsk, GN-24]A-6 {1968). (z) D.E.T.F. ASH~Y and J. W. M. PAUL: IV International Con]erence on Ionization Phenomena in Gases, Vol. 4 A (Uppsala, 1959), p. 961. (~) R. G. GUDDAr Thesis, Oxford (Culham Lab. Rep. CLM-M-52, 1966) {1967). (4) Yu. A. KOLES•IKOV, 1W.V. FILIPt'OV and T. T. FILIPrOV: Kurehatov Report 18/904 (Culham CTO-8) (1966). (b) E. ])UGH and R. PATI~ICK: Phys..Fluids, 12, 2579 (1967). 41

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IL NUOVO CIMENTO VOL. 27 B, N. 1 11 ~aggio 1975

Propagation of MHD Shock Waves into Regions of Various Uniform Densities.

S. CHA~DRA and K. M. SRIVASTAVA (*)

Department o/ Mathematics, University o/ Jodhpur - Jodhpur

(ricevuto il 28 Ottobre 1974)

Summary. - - In the presence of uniform magnetic field in an infinitely conducting fluid, the pressure behind the MHD shock wave is investigated after the shock wave has interacted with the density variation parallel to the shock front. The case is generalized to another density variation perpendicular to the shock front. The density variation is due to a small discontinuity jump across the interfaces. The problem is formulated under the assumption of small change in density and the shock profile is obtained. It is observed that the curvature of the shock increases as the Mach number decreases or the strength of the magnetic field increases.

1. - In troduct ion .

The interest in plasma shock waves was s t imulated by the possibility of

heat ing ions rapidly to thermonuclear temperatures. Lately, the interest in

shock waves has centred on geophysical and astrophysical phenomena. I n

the laboratory, the pinch effect is generally used to compress an initial plasma,

thereby generat ing a cylindrically imploding shock. Iqoneylindrical pinches occur

in very short pinch tubes (1), toroidal pinches (2), plasma guns (8), and plasma

focussing devices (4). Simulation of geophysical shocks has also been per-

formed in a plasma wind tunnel driven by an are source (% Satellite observa-

(*) Present address: Institut fiir Plasmaphysik der K.F.A., Jiilich. (1) A. E. ROBSON and J. Sn]~FFI~LD: I.A.E.A. Novisibirsk, GN-24]A-6 {1968). (z) D . E . T . F . ASH~Y and J. W. M. PAUL: I V International Con]erence on Ionization Phenomena in Gases, Vol. 4 A (Uppsala, 1959), p. 961. (~) R. G. GUDDAr Thesis, Oxford (Culham Lab. Rep. CLM-M-52, 1966) {1967). (4) Yu. A. KOLES•IKOV, 1W. V. FILIPt'OV and T. T. FILIPrOV: Kurehatov Report 18/904 (Culham CTO-8) (1966). (b) E. ])UGH and R. PATI~ICK: Phys..Fluids, 12, 2579 (1967).

41

~ 2 ~. ('IIAN1)IC~ 3 l i d K. 31. SHI;ASTAVA

t ions have confirmed the presence of collisionless bow shock (~), which pro tec ts

the magnetosphere fl 'om lhe Snl>ersonie flow of 1)lasma f rom the Sun. These

satell i te observat ions have h,d to a large m tmbe r of theoret ical papers on shock phenomena .

N H I ) shocks have been classified in lhe one-fluhl model b y the na ture of the jmnp eo~tditions wit hont l'eferem, e lo the stru(,tm'e (7). The shock s t ruc ture prob lem reqnires lhe solution of lwo-flnhl eq / t a ! io l l s . Complete analyt ica l solu- tions have been possible only for very weak shocks propaR'ating perpendicular

to the mag'netie fiehl (s). StronR' shock sohgions for very simple eases have

been obta ined with no lhermoeleetri<, effecls (,,.~0). CHESTER all~l COLLINS (I1) NI[VO derived the shape of the shock front aft, er

in teract ion with lhe densi ly ehan~'e il~ ordinary R'a~; usinlz the me thod devel-

oped b y LmHTm~A~ (1=') and used by Cm-sTmt (~3). The anthors have used lhe meti~od of CI[ESTEI~ a Ild (?OLL1NS (H) in the s~udy

of propaga t ion of MHD shock wave in a per fec t ly <,onduetinR" p lasma in which var ia t ions ill den.sity are subjected to at),:ul>t etmn~es across the interfaces

A : ~ 0 and 1 " = 0 along the l)ositive X-axis (;nly, i.e. in the reo'ions i) :Y < 0, ii) A ' > 0 , Y > 0 and iii) A:>-0, ) '<20 . X ' a n d Ybeino" Cartesian axes par'ri- le! lo the du'cet on of pl 'opao'ation of sho<.k wave and uni form mag'netie fiehl

respeelivcly.

2. - MHI) shock transition relations.

Iaet: U - - (U, 0, (I) be the uni form speed ,,>f shock, 0, 9o, Po, a0, B ~ (0, 0~ B) and q ~ = ( ( h , O, 0), 9~, P l , a~, B ~ - - (0, O, B~) be lhe velocity, density, pressure, speed of sound and Mreno'th or' mao'netie field ahead of and belfind the shock.

Then M H D ,~hoek tr:m.~ition relal ions <'an be wr i t ten as

1) ~(,U -- ~l(U--q~) ,

('-'~ ~ t "~ l - - U u q ' ! l,',, 1 1 ~ o

(3) ':~ I1 - ( U [ q,)'-] ;:p,~ (1 P~'2o~ B~ (1 /:FI''~

(4) B o x U B~/,(U--qr

(") N . F . N:l.:s~'. ('. S. ScI.:.~I:<'l: trod J. II. S::l,:l<: ,l,ur~t. (h~ophy., ". Res., 69. 353l (1964). (7) J . ]';. ANDI~;t~SO_'<: M i l l ) ,%o,'/," Ware.,' (('aml>rid~'c. 3lass.. 19(i3), (s) P. N. l i t / : Phgs. J,'luidx. 9. S9 (1~,~6!. ('J) W. ) i : ~ s H a l , : Pro,. t~oy. X,,... A 233. ?,U7 (1t}55). (lO) \V. (~H;Cl:!a t[. J . KA!-:PP~:I!;V and il. MAYsE]t: _Vt~cl. F~t.~'io~t S~.ppl..2. 403 (1962). (u) 1V. ('ul's~II-:~ and II. ~'~mll:<~: l.~racl .lot~r,. 7"tdt~t.. 8. 345 (1970). (12) 5[. J . I,IGiUIIIII,L: ]'FOe'. Ilo.i/. Not, . . . \ I98, 454 (1949). (la) ~V. ( ' l lLs'rM:: (d~u~r:. J,',~:;'n. 31c~b. ,I/qd. 3[a/h., 7, 57 (1954).

PROPAGATION OF MIlD SHOCK WAVES INTO REGIONS ETC, ~

I t is convenient to introduce Xo, Yo, Q and M, as

(5) ~o~ _ U B1 __ Xo , P~ =- Y o , ~o U-- ql Bo Po

Q ~ B~/2,upo the r~t, io of magnetic pressure and hydrostat ic pressure, grid

== + + / I.tt~o ~o J

M , and M being magnetic Maeh number and Much number in ordinary gas respectively.

Solving these equations, as in FERRARO and s (~4), we obtain

(6)

(7)

where

and

X o = -- C- - DM2, d- (C~'d - EMS, d- FM,~) t ,

Yo ( 2 Q + y ) M ~ : ( I - 1 ) - Q ( X : - I ) + I

c 2z(Q+~) D (y --1)(20 + Z) 4q(2--Z) ' 4q(2--y) '

E - - (_Q(S -F 2 y ~ 2 y 2) + 2y + 3y2--y3)Q + y 3 _ y ~

])2.

I : [ c t i c e

(s)

Also

(9)

Ot--o~Xo, B1 BoXo, q t = U(1-- , :~'o) and p t = p o Y o .

3. - R e f l e c t i o n a n d t r a n s m i s s i o n o f M H D s h o c k w a v e a t a n i n t e r f a c e .

We consider the interaction of MIID shock w~ve with an interface (parallel to the shock front), which separates the two regions at the same total pres- sure, and uniform, but different densily given by the relation a~= c o ( l + s). The notat ion for the regions is same as tha t used by CHESTER and COLLI~S (1~). The velocities of the reflected 9rod t ransmi t ted shock are denoted by

(14) V. C. A. FJ.mi~i~o and (!. PLt~,iI"roN: An I~,~troduction to Magneto]Iuidomechanics (Oxford, 1966').

44 S. ( ' I I A N I ) I G k ; l l i d K. M. SRI V A S ' I 'A V A

a~--q~[1 q - ( Y + 1)b~'/2] and a . , M , ( [ - - G o ) respectively. I f we assume tha t lei <<1, the flow quanti t ies behind the reflected and t ransmi t ted shocks a.re deduced f rom relations (8) with appropria te notations and modifieations:

( 1 0 )

( ~ ] )

(12)

and

(13)

(~)

(~5)

qs : a2M,(l - - b~s)"

[ 1 . ~ . , , ~ ~ ~.lI, ! '~ , )~-- b2~ �9 1 Xo- - . . . . . ~-o.-V~r162 '-7 EM2, i I"M4,) �89 J

Pa - - Po [(2Q ~- x)M~(] --2b,2v).

{ 1 2DM~*(('2-- , ;. * , . i , ) 4 } , E M , . t M , ) - ' - - ( E 3 , - - z F M , ) �9 1 Xo i X~((--?+ 3 ~ - - ' 3 ' � 8 9 " b,2~ - -

~ "" 2D3I:((;~'-- E M : +-FM:)I--(]~'M:d-2"~M'*) b2~)-~-1]

�9 - ( ( ' ~ : J~M~ - .I;'M++,) �89 b+++:,

% = % { 1 + (y + 1)b~-12},

P4 = p~{1 -- (2r 2 _L y) M,2(y __ 1)q,b~,/2a~ + Q(F + l )q~b~s/a,} ,

B 4 = BoXo{l -- (7 -',- ] )q~ble/2a,} .

Since veloci ty and pressure are continuous across the interface we obtMn from eqs. (10)-(15)

(16)

where

~, <-9~ y) M,(- - ~) l l

. (2r - 7) ;P,(2 + ,';) 1- ' b . : 1 : - • '

S = 4 ) 2 C 2 .EM", ' ~ , 4 , � 8 9 EM~, -- 2 F M , - - 2DM,( ~ ~- ~_, ~,~.~

Xo(Xo - - 1.)(C ~ + E M , ~ + FM~,)

4 . - D i f f r a c t i o n o f M H D s h o c k w a v e a t a j u n c t i o n o f d i f f e r e n t d e n s i t i e s .

To generalize the ease the region 2 is divided into two regions denoted by 2u and 21 by the plane interface perpendicular to the first interh~ee,

The mathemat ica l formulat ion to obtain the flow variables in the region bounded by reflected and t r ansmi t t ed wave is the same as tha t given by CHESTER and COLLINS (n) ~nd developed by LmHTmLL (~2).

PROPAGATION OF MltD SHOCK WAVES INTO REGIONS ETC. 4 5

Let X , 7s be the Cartesian co-ordinates with X-axis ~long the interface perpendicular to the shock front and Y-axis along the interface parallel to the shock front and let us take illitially the junction of these interfaces as origin. Let V, P, R and B denote respectively the velocity, pressure, densi ty and magnetic field in the diffracted wave. Then the equations of motion are

DR (18a) - - + RV. V-- 0 ,

Dt

(18b) R D V ~ / + V(P § B2/2#) : 0 ,

(18c) D CP ~ B2/2/~ __ 0. Dt \ R y ]

F rom eqs. (18a) and (18o) we obtain

D P (19) Dt + r ( P ~- B 2 / 2 / ~ ) V ' V 0 .

Since the flow variables V, P , R and B differ by small quantit ies from their values (q~, 0), Pt, ~1 and B1 in the uniform region behind the shock before dif- fraction, eqs. (18b) and (19) can be linearized as

DV D P (20) ~ § V(P +B~/2#) 0 and -~--/~- ~(P~ -/B2,/2#) ~-- 0 .

We consider the transformation

X - - qlt (21a) -- x ,

a x t

Y (21b) -- y ,

alt

(21e) V-- q1(1 ~- uv) ,

(21d) P + B 2 / 2 # - - P ~ - - B~/2tt = p* ,

a1~)1 ql

and we assume a more nearly exact equation of the t ransmi t ted shock in the new transformed plane (x, y) as

(22) X : Ut{1 + ](y)} or x : k + U](y)/al ,

whose normal velocity is given by V~ = [1 ~- (]-- y]'), u/'lal].

46 S. C I ! \ N D [ ( \ i l l i ( [ ix. "q. > l l i V \ ' r

rJ?he 'shock l runs i l ion l'lqul ions 1,311 l)e deduced ill torlns (if no1'1113] ve loe i ly a~

[' (- " l . \ a 2 / \ 'r J ] '

, {C-' ~ ~ t " " l ' \""-I I )

--r - - ( ' / ) ~ " ('!"-: E\a~/ ' \a~/ j j

(25) B :== B0 - - c-/) t,,,-~/ ~ ~ "

Front eqs. (21c) and (~3) v,e ~'('i

(26) ~ = (1 47 ,~)(/- ,#') 47 ,~'~-~,

(27) ~, = -- kX,,l' ,

and f rom eqs. (21d), (24) an(t (25) we get

(28) p * - - /,.(2-# , ~ ' ) ( / - y i ' - e ) ,

where p* is the to ta l pressure.

These b o u n d a r y eondil ions ha ~e lo be imposed on x = lc. Ill eq~. (26), (27) and (28) the a l )propr ia te valu(~ of ~: defined in t e rms of the u p s t r e a m region has 1)een used.

5, - Boundary-value problem for the pressure.

F r o m eqs. (20), (21a)- ( 'qd) , (26), (27) a nd (28) we obl.qin

(29)

where

(30)

]: (k 8p* 6p*~ ~p* 8p* t t k 6p* &~' "" 6~,! ] c,c ?y y 8y \

1--, '; Xo A - H -

k(2 i-,~') ' 2 4 S"

The r ema in ing bourM~u'y ( 'ondil ions on xa47 y - ' = I , x < k are

(31) P* I -- 2bleu '

/ 2blez ,

y > 0 ,

y < 0 .

PROPAGATION 0]" MIID $IIOCK ~VAVI;S INTO REGIONS ETC. ~ 7

This gives a d iscont inui ty of magni tude --2ba(e.--s~) in p* along the

b o u n d a r y a t x = - 1, y = 0.

There is ano ther d iscont inui ty of pressure p* along x -= k and y = 0

whence f rom eq. (28)

p* = # ( 2 + ,S)(/-- y / ' - - e~) , y > 0 ,

(32) p* = k(2 + S)(] -- y]' -- ~ ) , y < O,

which gives a d iscont inui ty along x ~ k of magn i tude k(2 q- S)(s. - - eL) a t y = 0. I f we use the Busemaun t r ans fo rmat ion and the conformal t r ans fo rmat ion

as considered b y CKES~E~ and CoLLI~s (n), the bounda ry condit ion (29) is

t r ans fo rmed into

(33) 8p* /~p* 1 [ Hk ]

which holds for x~ ~ 1.

I f we consider

(34) W(z~)-- ~p- ~P": ~y~ + i ~x~

on y ~ = 0 , - - c ~ < x ~ < - - l , W ( Z l ) is real and f rom eqs. (31) near z 1 ~ - - 1

W 2bl(e.--e,) z (z~+ 1) '

and f rom eqs. (32) near z 1 = 1

W, k(2 + S)(s~--s,) n ( z ~ - - l )

F r o m eq. (33) on y ~ = 0, x~> 1

(35)

where

(36)

arg W(zJ = ~g-1 C2(xi - - 1)

~k((x~ + 1)/(1 - - k'~)) - - A ( x l - - 1)

(x l - -1)~ (x l - -1)~ -----tg-' ~ / ~ + t g -1 V~ fl ,

1 [ (2+S)k(l--k ~)

~({ (S+~')k(1--k~)/~ (1 + s ) ( 1 ~ # ~) ~ k ~ x o "

4 8 5, ( ' I I A ~ D R , k il l l([ K. M. SR1V A ST A V A

The func t ion which satisfies these condi t ions and which is eve rywhere integ-

t ab le excep t a t z l = 4 1 is

(37) W(zl)=~ (.~fP~z--i~/z~--i) ~(X"~fl--i~/z:--t) --~.

�9 I ~'('~ + ">-2619 Z~)(: + ~!/(~,-~,). [~(zl - - 1 ) ;~(z 1 -ff .1) ]

F r o m eq. (34) it follows theft p* is t he im,~gin,~ry p~r t of fW(zl)dz~. Thus

[ 2/h(zc'-fl) t~, ~ /k.~/ /~ (3S) p* /~'(2 ;~)(~-,,--e,) ,,- - . - = . . . . . • .~ k(2~ S ) ( c ~ - - l ) ( f l - - l ) . . . . . . . . . . .

1 ~ 2b,~(1 ,-fi) f i ] tg ' - - :'/o-.=- + i - o<~l; [(o<- iis<(~ --;,~)- ~V~, ~-?z-

1 ( /3- - 1)1, (_ -; ,~)J (g ' - ' sgn y - - b,(r~ - - ~,)

whe re t he cons t~n t of i n teg ra t io ]~ is such tha.t, p*-)-2b~e,, as y~--> co. E l i m i n a t i n g p* fi'oi~1 eqs. (28) a, n4 (38) :rod integT~%ting we obtMn the shape

of shock as

~( _ _ _ _ , 2 ~ (39) /(y) : : ~ 1 /,.(2 =- N)] [(c,, ~ ~ ) - (e,~--~-,)g(y,.I,")},

0.8

06

O.&

0.2 Y

1.0'

1 I I l I )

0 0.2 0,4+ 0.6 0.8 ] .0 y /k r

Fig. 1. - 5tI[.D shock profile, if(y/l,") for M = 5 and Q ~: (),1, i. 4, 8 and 16,

PROPAGATION OF MHD SIIOCK WAVES INTO REGIONS ETC. ~9

where

~ ( 2b~ ~)) g(y/k') = (40) ~ 1 k(2 - -

2b,(~+fi) Y ] I 2b, a(l+fl) } k(2+S)(a--1)(fl---iitg l V / k , ~ _ y 2 ~L a - - f i t ( r fl "

y _~_ 1 { 2b1~(1 --~ ~\ ~ y " tg-~V/#7~_y,~ ~ a-- (fi--1)k(2 § s)j tg<fl~/k'~--y~ §

/- k ~ ~ ( r Jr- ,S) - f l tg-'{(~2--1)(1--y2/k")}�89

+ k' fl(zt--fl) ~--(fl S) tg-'((fl2--1)(1--y2/k'~)}~+

+

T h e shock is c o n t i n u o u s a t y = 4- k ' s ince t h e u n d i s t u r b e d p a r t of t h e

t r a n s m i t t e d shock w i t h t h e a p p r o p r i a t e v a l u e of s is g i v e n b y

2bl (41) X=Ut{I+(1- k(2+S))S}.

1.0

0.8

0.6

OA-

0.2

- :10

I I I I I ~, 0 0.2 O.h 0.6 0.8 1.0

y /k r

Fig. 2. - MHD shock profile g(y/k') for Q = 1 and M = 1.5, 3, 5, 10 and 20.

4: - I I N u o v o C i m e n t o B .

5 0 ~. ( 'IIANI)RA 3 l i d 1,~. 3I. 5RIVAS'I'~VA

:Near y : k'

y ) 2k(2 ~ S) (42) g F := 1 -!.- ~-i~,(~ !: ,',') ~ ~b,]

other h igher -o rder t e r m s .

E q u a t i o n (42) impl ies t h a t c u r v a t u r e in the shape of the shock is con t inuous .

We p r e sen t the shock profiles in Fi~'. I a n d 2. Fi,~'ure I r epresen t s the shock

profiles for Q(-~ B~/2/zpo)--1 a n d dif ferent va lues of 3I , the Maeh n u m b e r .

F i g u r e 2 shows the shock profiles for 3 l 5 a n d d i f fe rent va lues of Q. I t is

seen t h a t c u r v a t u r e in the shoek proli le decreases as M increases whi le i t in-

creases w i th Q. $ $ $

The au tho r s are i n d e b t e d to Prof. KUSHWAHA for encouragement . . One

of t he au tho r s (K.M.S.) g r a t e fu l l y acknowledges the ,~upport of the H u m b o l d t

F o u n d a t i o n . T h a n k s ~re also due to the Di ree lo r s of I P P / K F A J i i l i eh for

p r o v i d i n g the f~eil i t ies a n d e n e o u r a g e m e n L

�9 R I A S S U N T O (*)

In presenzn di un campo magnetico uniforme in mt tluido infinitamente conduttore, si indaga sul l 'andamento della pressionc nell 'onda du r t o 3 [ l i d dop() the l 'onda d'ur~o ha interagito con la variazione di densit~ parallcla al fronte d'urto. Si generalizza il easo a un 'a l t ra variazione di dettsit'~ perpendieolare "d fronte d'urto. La variazione di den- sit~ g dovuta a un piccolo salto di diseontinuit5 fra le interfaeee. Si formula il pro- blema supponendo una pieeola variazione di densit'~ e si ottiene il profilo d'urto. Si osserva ehe la eurvatura del fronte d 'ur to aumenta al diminuire del numero di 3[aeh o a l l 'aumentare dell'intensiff~ del eampo magnetieo.

(*) Traduzio~e a eura della Redazione.

PacnpocrpaHeHne M HD y~apuo~ 8 o a u ~ B o6.~lacTn c pa3aw~nb~,m noc~oaam,~u

IIYIOTH0CTHMH.

ee3mMe (*). - - IJpH HaYlHqHH HOCTO,~IHHOFO MarH~rHoro rtonn B 6ecKoHenao npoBo~nule~ mI4~KOCT~t 14ccJJe~lyeTca }laByIeHHe 3a MHD y~IapHO~ BOnHOfi, I~or}la yztapHaa BOJ1Ha B3al4- MO~e-~CTByeT C I'IJ/OTHOCTbtO, H3MeRnrotaeiicn napan~eJ~bHO c~powry ynapno~ ~o~nsI. PaCCMOTpertHNfi cJ~yua~ o6o6taaexcn Ha Apyro~ c~y,m~q, gor~a IT5IOTHOCTB g3Meg~eYc~ rIeprIeNjIHKynapHo ~poaxy yAapHofi BOJIHb[. ld3MegeHHe IIj1OTHOCTtI o6ycJtoBJ~eno Ma- JIeHbKHM cKaqKOM *Iepe3 HoBepKHOCTH, I / Ic l lOJlb3yg ripe~lnoJlo?KeHRe o Ma.rlOCTld H3MeHeH~4g rInOTaOCrg, ogpe~enneTca IlpoqbH21b yAapHofi BOflHbl. Ua~JltO~aeTC~l, qTO KpHBIJ3Ha

yAapno~ BOJIHb[ yBenaaHBaercn, ~or;Ia qgcao Maxa yMeHbiJ-raexca ria~ tlHTeHCHBHOCTb MaFHHTHOFO IIOTlIt yBeJl l tq l IBaeTcI t .

(~ IlepeaeOeno peOaK4ueS.