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14
Вестник КРАУНЦ. Физ.-мат. науки. 2016. № 3(14). C. 29-33. ISSN 2079-6641
DOI: 10.18454/2079-6641-2016-14-3-29-33
MSC 76W05, 86A25
ABOUT PAINLEVE PROPERTY OF A HYDRODYNAMIC SYSTEM
G.M. Vodinchar12, D.S. Noshchenko12, A.S.
Perezhogin1,2
1 Institute of Cosmophysical Researches and Radio Wave Propagation FEB RAS, 684034, Kamchatsky Kray, Paratunka, Mirnaya st., 7, Russia
2 Vitus Bering Kamchatka State University,
683031, Petropavlovsk-Kamchatsky, Pogranichnaya st., 4, Russia E-mail: [email protected]
We represent conditions of hydrodynamic system when it passes the Painleve test. We use Kovaleskaya-Gambie method for fourth order ordinary differential system. We obtain Lorenz-like dynamic, hydrodynamic system.
Key words: Painleve test, Lorenz-like dynamic, hydrodynamic system.
© Vodinchar G.M., Noshchenko D.S., Perezhogin A.S., 2016
УДК 517.956
0 СВОЙСТВЕ ПЕНЛЕВЕ ГИДРОДИНАМИЧЕСКОЙ
СИСТЕМЫ
Г. М. Водинчар1,2, Д. С. Нощенко1,2, А. С. Пережогин1,2
1 Институт космофизических исследований и распространения радиоволн ДВО РАН, 684034, Камчатский край, п. Паратунка, ул. Мирная, 7
2 Камчатский государственный университет имени Витуса Беринга,
683032, г. Петропавловск-Камчатский, ул. Пограничная, 4 E-mail: [email protected]
В статье получены соотношения между коэффициентами системы гидродинамического типа, при которых система проходит тест Пенлеве. Использован метод Ковалевской-Гамбье для уравнения четвертого порядка. Рассмотрен пример движения в системе аналогичный системе Лоренца.
Ключевые слова: тест Пенелеве, система Лоренца, гидродинамическая система (с) Водинчар Г. М., Нощенко Д. С., Пережогин А. С., 2016
ISSN 2079-6641
Vodinchar G.M., Noshchenko D.S., Perezhogin A. S.
Introduction
The existence of magnetic fields of planets, stars and galaxy is explained by a dynamo-mechanism [1]. Mathematical aspects of dynamo effect are reduced to solutions of MHD equations. Nonlinear terms of these equations don't allow to find an analytical solutions (except private cases). Direct number simulation of 3D magnetohydrodynamics with big Reynolds number require huge computing at supercomputers [2]. So simplified MHD systems are very important to investigate for understanding main features of dynamo.
In the paper we discuss the analytical properties of a dynamical system, which is the simplest model of the dynamo. We find relation between coefficients of system in order to pass the Painleve test.
We discuss a model of process which generates the average magnetic field by a turbulence flow of viscous incompressible fluid with the alpha effect in rotating coordinate system. It is described by magnetohydrodynamics equations.
dtv + Rm (vV) v = PmAv - Vp - E-1Pm (ez x v) + rotB x B,
dtB = Rmrot (v x B) + Rarot (aB) + AB,
(1)
V ■ v = 0,
V ■ B = 0,
where v - the average velocity, B - the average magnetic field, p - pressure, f - mass density of external forces, a - tensor of the a-effect, Rm - magnetic Reynolds number, E - Ekman number, Pm - Magnetic Prandtl number, Ra - amplitude of the a-effect, ez - the unit vector of axis of rotation.
We admit that field axially symmetric with respect to axis ez. Solenoidal fields v and B give the sum of the toroidal and poloidal components.
We represent the following decomposition of the velocity and magnetic field into the sum of time-dependent amplitudes and stationary poloidal (toroidal) fields products:
v = xi (t)vT (r)+ x2(t)vP (r),
(2)
B = yi(t)BT (r)+ y2(t)BP(r).
Substitution of decomposition (2) in (1) gives system of amplitude's equations [3]:
dxi _i
— = RmAii2XiX2 + E PmPi2X2 + Fi + RmLii2yiy2 - MiXl,
at
dx2 2 _i 2 2
— = RmA2iixf + E iPmP2ixi + F2 + RmL2iiy2 + RmL222y2 -
dy^i (3)
— = RmWii2xiy2 + RmWi2ix2yi + RaWia2y2 - Wi,
dy2
— = RmW222x2y2 + RaW2aiyi - ^2y2.
at
In the system we take into account that in the axially symmetric case the vector lines of any poloidal field lie in planes passing through the axis of rotation, and the lines of any toroidal field perpendicular to them. Uppercase letters denote constant
coefficients. They appear after application of Galerkin's method to a system. Coefficients U h n determine the dissipation rate of velocity and magnetic fields modes from (2). In this case we assume P12 = — P21 and some other relations on coefficients A112 = — A2n, L112 = -W112, L211 = — WI21, L222 = —W222, Ui = U2, ni = n2, F = 0.
System (3) is the simplest dynamo model without a kinematic effect. In this paper we investigate its analitical properties.
Simplified ODE system
We present the Painleve test on simplified ODE system. The Painleve test is neccesary condition for the Painleve property. The formal solution is given by Laurent series near a movable singularity x — x0. For this aim we use Kowalevski-Gambier method [4]. Main steps of Kowalevski-Gambier method are
1) substitution u(x) = u0xp in order to find integer value of parameter p.
2) computation coefficients u(x) = Wjxp+j for integer j. For every fixed j we have a linear algebraic system on coefficient uj of Laurent series. If a system is consistent we have an uniformal solution for fixed j. For some values of j system can be overdefined. In this case coefficients of Laurent series are free. In order to determine j for which system is overdetermined we can calculate Fuchs indices.
3) If a linear algebra system of Laurent series coefficients for each Fuchs indices is consistent the ODE system passes Painleve test. It does not imply the Painleve property. If system is inconsistent ODE system passes test.
Let us discuss one simplified case of (3). We suppose that L222 = L211 = W121 = W222 = F = 0, F2 = M, RaW2a 1 = Ra2 = a, P12 = K, P21 = —K, X1 = U1, X1 = U2, y1 = u3, y2 = u4, independent variable t = x. So a reduced ODE system is
U1 = —A U1 + Ku2 — LU3W4
u2 = —A u2 — Ku1 + M (4)
U3 = Lu1 W4 + a W4 — W3 U4 = a W3 — W4
where M,L,K, A, a - independent parameters.
1 step. We substitute ui(x) = wij0xpi,i = 1..4 at system (4). Main terms give next system on pi: p1 — 1 = p3 + p4,p2 — 1 = p1,p3 — 1 = p1 + p4,p4 — 1 = p3. Solution of the linear system is p1 = —2,p2 = —1,p3 = —2,p4 = —1.
We find coefficients ui,0 from system of main terms for calculated values of p;, i = 1..4:
p1x1 = —Lx3x4, p2x2 = —Kx1, p3 x3 = Lx1x4, p4x4 = a x3
From this system we get one trivial solution and four nontrivial
2 2K 21 21
u1,0 = -— ,u2,0 = -— ,u3,0 = ±7— ,u4,0 = ^ —,
La La La L
where 1 - the imagery unit.
21 21
Let us check the first solution u30 = —, u40 = —, when u30, u40 have the same
' La ' L ' '
sign. In this case Fuchs indices are irrational 0 , 1 ,5/2 — 1/2^17,5/2 + 1/2vT7. Because of the irrational value we need to take u3,0, u4,0 with the different sign. For positive
ISSN 2079-6641
Vodinchar G. M., Noshchenko D.S., Perezhogin A.S.
Puc. 1. One velocity and two magnetic cmponents
values of ^3,0 and ^4,0 the Fuchs indices are -1, 1, 2, 4. For each Fuchs index we get 3 invariants:
8(A-1) K2 + A 2 + 4 A - 4 Q1 =-;—, Q2 = -2-
a L
La
Q (n 2) ) K + 4 (A - 1) (a 2A - A3 - 2 A + 2) + A (3 A - 2)(-u2,iA + M - uw) Q4 = ((A - 2) U2,1 - U2,2) K + 4----h
La
K
Assuming Qi = Q2 = Ö4 = 0 we get next set of system's parameters A = 1, K = ±1, M = 0.
General ODE system
In this case we assume P12 = —P21 and some other relations on coefficients A112 -A211 = A, L112 = —W112 = —L, L211 = —W121 = —P, L222 = —W222 = —Q, M1 = M2 = n1 772 = 1, F = 0. For such coefficients system is more general than previos one.
u'i = Aui U2 + Ku2 - Lu3 U4 - Ui u'2 = -Au2 - Kui - Pu2 - Qu4 + M - U2 u'3 = Lui U4 + PU2U3 + a U4 - U3 "4 = Qu2u3 + a u3 - u4
(5)
Let us consider the Painleve test of system (5) with A = P. First we check if the pivot terms satisfy following equations
P1 - 1 = P1 + P2, P1 - 1 = P3 + P4, P2 - 1 = 2 pi, P2 - 1 = 2 p3, P2 - 1 = 2 p4, P3 - 1 = P1 + P4, P3 - 1 = P2 + P3, P4 - 1 = P2 + P3
There is one case when p1 = -1, P2 = -1, P3 = -1, P4 = -1. Now we substitute expantions u; = "0,;xP; into pivot terms and get nonlinear system for coefficients "0,;.
t /l+v -l2-4 pq "0,1 = /"0,3, "0,2 = (LQU0,1 - P) , "0,3 = --
uo,4 = -QUo,2UO,3
In order to simplify "0,; we put Q = -L2/(4P). Such simplification gives a very simple characteristic equation for the Fuchs indices j(j3 - 6 j2 + 5 j + 12). Solving equation we obtain indicies -1,0,3,4. A characteristic polynomial doesn't depend on parameters L,P. There are two invariants for this system. Invariants correspond to the Fuchs values j = 3 and j = 4. Detailed analysis of the invariants gives some relations on coefficients.
So L = 2P, P = 1. Free parameters are K, a and dependent one is M. Resulting system passes the Painleve test with a = 3, K = 2.
If we put value of M to zero, we get a modification of (6). Direct solutions of this system gives periodic oscillations (fig. 1). If we keep constant force M, the periodic solutions will be relaxation oscillations.
Conclusion
We have considered two cases of hydrodynamic system. For these systems we find sets of parameters to satisfy the Painleve test. It is possible to check what conservation law holds for this system.
References
[1] Zel'dovich Ya. B., Ruzmaikin A. A., Sokolov D. D., Magnetic fields in astrophysics, RHD, Moscow-Izhevsk, 2006, 384 pp., [Zeldovich Ya. B., Ruzmaykin A. A., Sokolov D. D. Magnitnye polya v astrofizike. - M.-Izhevsk: RKhD, 2006, 384 (In Russian)].
[2] Frick P. G., Turbulence: approaches and models, RHD, Moscow-Izhevsk, 2010, 332 pp., [Frik P. G. Turbulentnost': podkhody i modeli. M.-Izhevsk: RKhD, 2010. 332 (In Russian)]..
[3] Gledzer E. B., Dolzhansky F. V., Obukhov A. M., Systems of hydrodynamic type and their application, Nauka, Moscow, 2010, 368 pp., [Gledzer E. B., Dolzhanskiy F. V., Obukhov A.M. Sistemy gidrodinamicheskogo tipa i ikh primenenie. Moskva, Nauka, 2010, 368 (in Russian)].
[4] Conte R. M., Musette M., The Painleve Handbook, Springer, Netherlands, 2008.
Поступила в редакцию / Original article submitted: 11.08.2016
Ml = Ml M2 — Ml + 2М2 — 2 M3M4, m2 = — M2 — m2 — 2m1 — m3 + m4 +
M3 = 2 Ml M4 + M2M3 + ЗМ4 — M3, M4 = — M2M3 + 3M3 — M4
1330 99 ,
(6)
