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25.00.10 { \, " - { 2011 .

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1, [1, 9], , . 1-3 , = k kc k kc , k:

R = Rc(k) + "2 ( ").

, , , ( ). , P Ta , . , . , "2=5 ( , L0 , , , ). , .

, , { ( ). , max("8=5; (k kc)2). , { , .

, . LW = W, L , , (x y ). { , { . , , , .

, A L ", . L0 ( ; ; 0) (k ; ; ); x y x y ( ; k x ; ). y A ". ( ) A x y ". :

(i) "4=5, 2 = P(2 k2 + 3Pk2) 1 y 2 4 3k2 = = C"2;

x y y ~ C { , P Ta. , = C"8=5, 5 ~ C = P(2 k2 + 3Pk2) 1C4=5.

(ii) "4=5, 12P k = 2 k2 + 3Pk 2k + = 2 k:

x y .

(iii) "4=5, :

"8=5:

( ) O("2). O("8=5) , "8=5 . , .

, " { , , , .

2, [9], , . , . "2 , , .

, , :

< P 0:677. , LW = W, L ( ), , , ! ( ~ x y !). { , { .

, A L ", . L0 = @=@t + L0 ( ; ; 0) x y !, { ~ (k ; ; ) ! !. ~ x y . A ". (, !) ~ x y Re( ), { max max A, ". :

(i) "2=Re( ) "4=3:

max .

(ii) "2=3.

Re( ) = C k2;

max C { , P Ta. 2k + = 2 k:

x y .

(iii) "2=3. Re( ) "4=3:

max ( ) O("2) "2. O("4=3) , "4=3 .

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3 , . [12].

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(SV, OSV ZZ, ) . 3 : SV2, < < 0:543 P 0:782. ", . , , . , , .

, 1, A = fAijg L , ". "2. det A > 0 (1) S(A)tr A det A > 0; (2) S(A) = A11A22 A12A21 + A11A33 A13A31 + A22A33 A23A32:

, " ; A ",, . , " x y , , (1) (2), .

x y , .

x y (k; R), , . 1 P. , , ; , . , . , (SV, skew-varicose) , , (OSV, oscillatory skew-varicose) { ; . (ZZ, zigzag) x y = 0, .

x < , P 0:782, = 0, y x ; . , SV2.

(1) (2) , :

12 SV : "2 > f1 + O( ); f1 = k; (3) OSV : "2 < f2 + O( ); (4) 108(P + 1)2 k f2 = ;

(P + 3)(3P2 + 2P + 2) + 3P2(P + 1)1=2(P + 5)1=9 P2 ZZ : "2 < f3 + O( ); f3 = : (5) 2(P + 1) , limP !1 f3 = 1. (3), (4) (5) A.

SV2 ( P < P1, P1 0:782) SV2 : > "2h1 + "(P P1)h2; (6) h1 = 0:0012 h2 = 0:018 : (7) (6) , = ("; ), " = P P1, . (7) . , , (1) SVSV SVOSV OSV SV () () SV SV OSV OSV ZZ ZZ () () SV SV ZZ ZZ OSV () OSV () . 1: () (k; R), , , , P = 0:6 () P = 0:7 (), P = 2 () P = 7 (), P = 20 () P = 50 (). { , { SV OSV, { ZZ - { SV2. : k, : R.

, "2 1.

, 1 2, ", , . , (P2 2P 2)2 8P2(P + 1) 4(2k)1= = ; = ;

1 9 (P + 1)2P 3 (P + 1)1=4P = ; = :

3 (P + 1) 9 P , " P. , , . (E l .) x y " "4 none SV " "x y P < P1 SV2 ; " = "x y x "4 "2 > f1 SV " "x y P < P1 SV2 ; " = "x y x "2 < f2 OSV " Re( ) "x y 2 2 "4 "4=3 > 0 SV " 1=2 = " 1=x y 2 < 0, OSV " 1=2 Re( ) " 1=x y "4=3 "2 < f "2 < 0; ZZ = 0; " = x y "2 > f2 ; "2 + 4 "2 < f3 2 "4=3 none E-l 2k + = 2k = x y 4 , , . k = kc . , : ; ; , ( -). , , . ( , ; , .) [3, 12] .

; () . , . , ( , ), . .

. . k , P, Pm, = Ta1=2 Q . , R R 80Q = 6000 54072010 = 15k k () () . 2: ( ) ( ) () Q = 1000 = 0, 2000, 4000, 6000, 8000; () = 2000 Q = 0, 500, 750, 1000, 1500.

. , , , , Rm(k) Ro(k) ( ) k ( , ). Rm(k) Ro(k) k , , R k, Rc kc P, Pm, Q.

Rm(k) Q . 2. Q, Rm(k) . Q , Rm(k) ; ( k) k k. R R k k () () . 3: ( ) ( ) = 200, Q = 200 () q = 50, () q = 500. , { (P = 0:1 0:05, ).

Q, Rm (. . 2b), Q c ( ), Rm c Q. ( Rm c Q .) Ro(k) !o(k) Q, P q. (. . 3). , k, Q q , . , k, , . , , . , Ro c , . q, Ro c , .

{ P q Ro(k) . , k Ro(k), Ro(k) < Rm(k); k { k > kmin k < kmax, { Ro(k) Rm(k), , !c(k) = 0. , , Ro(k) kmin kmax . , (. . 3), kmax = 3:4, kmin = 1:7 (P = 0:1) kmax = 4:3 (P = 0:05).

(P; ) 0:001 P 10 1 1000, , .4- Q q. .4 . < P (P 0:677) . , P .

Q . .

, P (. . 4 q = 50), .

q Q. Q q , .

4 . ( ), , , . , ( ) , P P () () P P () () . 4: (P; ) () Q = 5 (), Q = () Q = 500 () q = 0:01 (), q = 3 (), q = 5 ( ), q = 50 (-). q = 5 .

( 0 =2) . . C2 z1 = ( + ajz1j2 + b1jz2j2)z1;

_ (8) z2 = ( + ajz2j2 + b2jz1j2)z2:

_ , a < 0 b1 a < 0; b2 a < 0 < =2. , , , , . .

, (8), R2, . , :

I. 0 < =max max(b1( ) a; b2( ) a) < .

II. 0 < =2 max max(b1( ) a; b2( ) a) > 0; max min(b1( ) a; b2( ) a) < 0:

, -.

, , . (8) .

III. 0 < =2 max min(b1( ) a; b2( ) a) > 0;

max min(b1( )b2( ) a2; b1( ) a; b2( ) a) < 0:

S3( ), , ( S3( ) .) , - S3( ), :

IIIa. S3( ) ; , .

IIIb. S3( ) . - , .

IV max min(b1( )b2( ) a2; b1( ) a; b2( ) a) > 0:

, . , , , .

(P; ) . 5. III IV > 800.

III , S3( ), =2, =2; , . , , -. . P = 0:69 P 5 140 500. P = 6:4 = 548. . 5. , .

, IIIa, , , .

(P; ), , . 6 Q q. q 1 q > 1. q 1 ( ) max (b1( ) a; b2( ) a) = 0, P. Q. q; P.

q > 1 q. (P 1) a = 0 max max(b1( ) a; b2( ) a) = 0, , q . : P , P .

5 , 4, { , , . , . - , , . , III IV II I P . 5: (P; ) . : . : a = 0, . : max min(b1( ) a; b2( ) a) = 0 . :

II-IV . .

III III IV IV II II I I P P () () III III IV IV II II II I II I I P P () () III III IV IV II II I I II II I I P P () () . 6: (P; ) , , Q = 50 q = 0:01 (), q = 2: (), q = 2:5 (), q = 2:51 (), q = 2:55 (), q = 5: (). , . 5.

, . , , 3 10, 10.

3 10 P 1 Pm 1. , { . 5 , 1010 2 104, , 1024 1010. , Ta ! 1 Q ! 1 , - . , , - . , (0:001 P 1, 10 Pm P) . , Pm 5-7 1%. P : , P 0.01 0.001 . Rc P Pm . 7, Tac P c Pm . 8.

5 P ! 0 Pm ! 0, . P ! 0 Pm ! 0, . , . , , Rc Rc Pm P () () . 7: Rc () () Ta = 4 106 Q = 100 ( ), 500 (), 2000 (): () P = 0:01 ( ), () Pm = 10 ( , .

Tac c P P () () . 8: Tac () () c Q = 500, Pm = 0:1 ( ), Pm = () Pm = 10 ( ).

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[1, 3, 5, 9, 11, 12]. , [7, 8]. [2, 4, 6, 8, 10, 13, 14, 15].

1. .. // . .

{ 2006. { N. 6. { 40{51.

2. .., .., .., .. // . { 2007. { N. 417. { 613-615.

3. .. // . . { 2009. { N. 4. { 29{39.

4. .., .., .., .., .. // . { 2010.

{ N. 433. { 341-345.

5. .. : // . . { 2011. { N. 5.

[http://arxiv.org/abs/1102.4092].

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p 8. Podvigina O.M., Ashwin P.B. The 1 : 2 mode interaction and heteroclinic networks in Boussinesq convection // Physica D { 2007. { Vol. 234. { P. 2348.

9. Podvigina O.M. Instability of ows near the onset of convection in a rotating layer with stress-free horizontal boundaries // Geophys. Astrophys. Fluid Dynam. { 2008. { Vol. 102. { P. 299-326.

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11. Podvigina O.M. On stability of rolls near the onset of convection in a layer with stress-free horizontal boundaries // Geophys. Astrophys. Fluid Dynam.

{ 2010. { Vol. 104. { P. 1{28.

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{ 023133.

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{ 359-396.

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