Analysis of Stationary Points and Bifurcations of a Dynamically Consistent Model of a Two-Dimensional Meandering Jet Текст научной статьи по специальности «Физика»

Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 1, pp. 49-58. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220802

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 76B65, 37J99

Analysis of Stationary Points and Bifurcations of a Dynamically Consistent Model of a Two-Dimensional Meandering Jet

A. A. Udalov, M. Yu. Uleysky, M. V. Budyansky

A dynamically consistent model of a meandering jet stream with two Rossby waves obtained using the law of conservation of potential vorticity is investigated. Stationary points are found in the phase space of advection equations and the type of their stability is determined analytically. All topologically different flow regimes and their bifurcations are found for the stationary model (taking into account only the first Rossby wave). The results can be used in the study of Lagrangian transport, mixing, and chaotic advection in problems of cross-frontal transport in geophysical flows with meandering jets.

Keywords: stationary points, separatrices reconnection, jet flow

1. Introduction

Meandering jet current is a fundamental structure in the laboratory and geophysical fluid (or atmosphere) flows. Strong oceanic and atmospheric jet currents separate water and air masses with different physical properties. The Kuroshio in the Pacific Ocean and the Gulf Stream in the Atlantic are examples of such currents [1, 2]. Recently, there has been a lot of interest in applying the ideas and methods of dynamical systems theory to the study of transport, mixing, and chaotic advection in meandering jet currents. Various mathematical models have been proposed for modeling such flows [3, 4]. There are two main approaches: kinematic and

Received April 25, 2022 Accepted July 08, 2022

This work was supported by the POI FEBRAS Program (State Task No. 121021700341-2).

Alexander A. Udalov udalov.aa@poi.dvo.ru Michael Yu. Uleysky uleysky@poi.dvo.ru Maxim V. Budyansky plaztic@poi.dvo.ru

Pacific Oceanological Institute of the Russian Academy of Sciences ul. Baltiyskaya 43, Vladivostok, 690041 Russia

dynamic. The first case is a generalization of experimental work, the result of which can be a streamfunction that determines the motion of passive tracers (particles that quickly accept the flow velocity and do not affect its properties) in this flow [3]. In the second approach, the streamfunction or the velocity field for passive tracers is determined taking into account the laws of fluid motion [5].

Observations of trajectories of buoys of neutral buoyancy in different regions of the Gulf Stream have shown their large-scale movement across the jet [6]. This behavior is considered as a strong argument for the assumption of chaotic advection (first proposed by H.Aref [7]) as a mechanism of large-scale mixing and transport in ocean and atmospheric jet currents (see [1,6]). In laboratory experiments [8-10] on imitation of jet streams in the ocean and atmosphere, the presence of chaotic mixing in a quasi-periodic flow on both sides of the jet was noted, but not transport across it. According to the widespread opinion, the large gradient of potential vorticity in the central region of the jet is a barrier that prevents transverse (cross-frontal) transport [5, 8-11]. The problems of the permeability of such barriers take a special place in oceanography because, while the physical and chemical properties of water on both sides of the jet may differ, the transverse transport from one side to the other can lead to adverse effects (for example, the situation with the spread across the Kuroshio of radioactive pollution formed after the accident at the Fukushima nuclear power plant in 2011 [4, 12, 13]).

For modeling the transport and mixing of water (air) masses, the Lagrangian approach is convenient, in which the motion of passive tracers is considered. The equation of motion of a passive particle is determined by classical mechanics:

£ =t), (i.i)

where r and v are the radius vector of a particle and its velocity at a point. The relationship between the Eulerian velocity field and the Lagrangian trajectories of tracers can be quite complex. Even a simple deterministic Eulerian field can generate unpredictable Lagrangian trajectories of particles. It is known from the theory of dynamical systems [14] that solutions of such equations can be chaotic in the sense of exponential sensitivity to small changes in initial conditions and/or parameters.

The present work is devoted to the study of the properties and structure of the phase space of the dynamically consistent model of a meandering jet. The conditions under which topologically different regimes are realized in the flow are found numerically: with heteroclinic and homoclinic types of separatrix connection. These results are planned to be used in further work on the study of the cross-frontal transport of passive tracers when taking into account the second Rossby wave and the external multiperiodic perturbation. Similar studies have been carried out in [3, 5], where the amplitude mechanism of breaking the central invariant curve and the emergence of a regime with cross-jet transport was discovered using the example of kinematic and dynamically consistent meandering jets. The analysis presented in this paper is an important step in understanding and choosing the values of control parameters in problems with complex (multi-period) perturbation.

In Section 2, a mathematical model of a geophysical flow with two propagating Rossby waves is introduced. The resulting system of advection equations is a Hamiltonian system with | degrees of freedom. The variables are changed in such a way as to get rid of the time dependence in one of the waves. In Sections 3 and 4, the position, number, and type of stability of stationary points that determine the structure of the phase space are studied. The analytical calculations presented in these sections are a standard technique for bifurcation analysis of low-dimensional

Hamiltonian systems. In Section 5, the results of the numerical solution of an algebraic equation are presented, which make it possible to construct a bifurcation curve separating the homoclinic and heteroclinic flow regimes. The last section discusses the results of the presented work.

2. Problem formulation and derivation of equations of motion

As a model of the jet stream in the ocean, we consider a two-dimensional flow satisfying the law of conservation of potential vorticity (Jj + v V) II = 0, which determines the dynamics of an incompressible fluid in a rotating coordinate system on the Earth's surface. The potential vorticity on the f-plane has the form n = V2^ + fy. The streamfunction ^ for such a problem is defined by the sum: the stationary zonal flow and its perturbation, which is represented as a superposition of running Rossby zonal waves [11]

* = *o(y) + E (yyk> (x-cjt] ■ (2-1)

j

Substituting the streamfunction from Eq. (2.1) into the law of conservation of potential vorticity and using the Bickley jet u0(y) = i/.max sech2 as the stationary zonal flow directed from west to east, we obtain a streamfunction satisfying the linearized equation of conservation of potential vorticity and consisting of a stationary jet and two propagating Rossby waves

^(.r, y, t) = —umaxL ^tanh — [Ax cos k^x — c^t) + A2 cos k2(x — c2t)] sech2 j-^j, (2.2)

where umax is the maximum flow velocity, L is the measure of its width, A1 2 are the amplitudes of Rossby waves, and k1 2 and c1 2 are the wave numbers and phase velocities of waves, respectively.

Let us connect the analytical form of the obtained streamfunction to the parameters of the laboratory experiments carried out in [8-10], where a looped jet was created with two waves whose integer lengths, n1 and n2, are fits on a circle of radius R.

The wavenumbers and phase velocities of Rossby waves may be calculated using the experiment's parameters [5]

mNV2 _Um^_Af2

I ' Cl-2~ 6E2

1 AT nn . . . . N-

K'2 = -rrcli2 = (2.3)

where A^i = d* and N.? = are coprime integers, i.e., ^r1 is an irreducible fraction.

1 1/ i 2 l/i N2

Passing to a frame of reference moving with the phase velocity of the first wave and using normalization,

x + c2t^r+Rt ,n ^ x ->•-—R, y —y Ly, t ->•-, (2.4)

m mumax

we obtain a streamfunction the evolution of which in time is determined only by the 2nd wave: ^(x, y, t) = - tanh y + sech2 y[A1 cos (N1x) + A2 cos (N2x + wt)j + C2y, (2.5)

where

= '2N'{ = 2N2(N*-N$)

°2"3 (N* + N*y 3 (Nf+Nl) ' 1 bj

and the advection equations have the form

dx

— = — C2 + sech2 y[ 1 + 2Al tanhy cos {N-^x) + 2A2 tanhy cos (N2x + uit)], ^ = - sech2 y[AlNl sin (A^.r) + A2N2 sin (N2x + cot)}.

(2.7)

The resulting streamfunction (2.5) has four control parameters: the amplitudes of two waves A1 2 and two numbers N1 and N2, expressing the relationship between the spatial periods of the waves. The numbers N1 and N2 are determined by four parameters of the experiment: umax, /3, L, R.

With this streamfunction it is possible to investigate the conditions of cross-jet transport for any pairs of numbers (N1, N2), the freedom of realization of which is provided by variation of the experiment parameters: radius R, jet width L, maximum speed umax and

3. Finding stationary points

The system of equations (2.7) is a Hamiltonian system with one and a half degrees of freedom. Chaotic advection is usually observed in such systems, which in the case of a small time-dependent perturbation is closely related to the separatrices of an unperturbed (stationary) system and to nonlinear resonances between the perturbation and the dynamics of the system around elliptical points. Thus, for a qualitative description of the chaotic advection of passive tracers, it is necessary to know the stationary points of the system, the type of stability, and the dependence of their number on the control parameters of the system.

Consider the system of equations (2.7) in the stationary case, i.e., when the amplitude of the second Rossby wave is zero (A2 = 0).

The equations of motion take the form

^(x, y) = — tanh y + A1 sech2 y cos (N1x) + C2y,

dx

— = P(x, y) = — C2 + sech2 y[ 1 + 2Al tanh y cos (A^a:)], (3 1)

dy

— = Q(x, y) = — NlAl sech2 y sin (A^.r),

and depend on three control parameters C2, A1, and N1. The parameters N1, N2 £ N*, and the parameter C2 G (0, |) and takes rational values determined by the expression (2.6). The parameter A1 e R, but since the system (3.1) has the symmetry x —> x + A1 —> —A1, we can assume that A1 > 0.

To find stationary points, it is necessary to equate the right-hand sides of equations (3.1) to zero. Then from the second equation, it is possible to determine the location of stationary points along the x axis:

kn ,

= —, fceZ.

By equating ^jf = 0 to zero, we obtain the equation for y:

sech2 y[1 + 2sA1 tanh y] — C2 = 0, s = cos N1x = cos kn = ±1. (3.2)

Let us introduce the variable z = sech2 y. Then tanhy = q\J 1 — z, where

' +1, y> 0,

q = ,

— 1, y< 0,

and Eq. (3.2) becomes

C2 — z = 2A1qszs/l - z, z £ (0, 1]. (3.3)

Squaring (3.3) gives a cubic algebraic equation with respect to the variable 2 (Fig. 1):

F (z) = 4Afz3 + (1 — 4Af)z2 — 2C2z + C22 = 0. (3.4)

Fig. 1. The graph of the cubic equation at the values of the parameters A1 = 1, C2 = 0.4

The roots of the cubic equation can be directly obtained using the Cardano formula, but they have a complex form, so to find out the position of stationary points and their number, we will conduct a qualitative analysis of the dependence of the algebraic equation (3.4) on the values of the control parameters. Let us determine the number of roots of the cubic equation (3.4) belonging to the range of permitted values 0 < z ^ 1. Since the coefficient 4A\ for z3 is positive and F(0) > 0, the first root, z1, is less than zero and lies outside the range of valid values. Since F(1) > 0 and F(C2) < 0, F(z) has two real roots z2 e (0, C2) and z3 e (C2, 1) (Fig. 1). From Eq. (3.3) it follows that the root z2 corresponds to the case qs = 1, and z3 — qs = — 1. Thus, the system (3.1) has the following stationary points:

x =

2k_ ÂV

y = arctanh sj 1 — z2

y = — arctanh sj 1 — z3

2k + 1 [ V = — arctanh sj 1 — z2

N

y = arctanh \Jl — z3

unstable point, stable point, unstable point, stable point.

(3.5)

The determination of the stability of the stationary points represented in the formula (3.5) will be shown in the next section.

4. Determination of the type of stability of stationary points

To determine the type of stability of stationary points (3.5), we write down the characteristic equation of the system (3.1) linearized near a stationary point

Px — \

P,

Qx Qy A

= A — Py Qx = 0,

Py = 2A1 sech4 y cos (N1x) — 2 sech2 y tanh y[2A1 cos (N1x) tanh y + 1] = = 2A1sz2 - 2qzy/l - z feAtfsy/1 - z + l), Qx = —AN sech2 y cos (N1 x) = —A1N'2sz.

(4.1)

x

A stationary point will be stable if PyQx < 0 and unstable if PyQx > 0. The stability condition has the form

G1(z) = A1(Sz-2)>G2(z)=sqy/T^z. (4.2)

Consider the case sq = 1. As shown above, in this case, the stationary point corresponds to the solution of Eq. (3.4) z2 E (0, C2). Since the function G1(z) is monotonically increasing, G2(z) is monotonically decreasing, and Gl (|) = 0 < G2 (|), the only root of the equation G1(z) = G2(z) is greater than |. Since z2 < C2 < |, it follows that. G1(z2) < G2(z2) and, therefore, at sq = 1, the stationary point is unstable.

Consider the second case sq = —1. In this case, the stationary point corresponds to the value z3. It is easy to show that G1 (z) > G2(z) if z > zc, where zc is the root of the equation G1(z) = G2 (z):

. _ -l + 12^- .yTTT2lf

18A2 * i4"^

Fig. 2. Function graphs G1 (z) and G2(z)

For A1 < 0.5 zc < 0. Therefore, for all admissible values z G (0, 1], G1 (z) > G2(z), and the stationary point is stable. Let us show that z3 > zc for A1 ^ 0.5.

Obviously, z3 > zm, where zm is the point of minimum of function F(z)

_ -1 + 4Aj + y/1 - 8Aj + 16A\ + 24AjC~

Zm ~ 12ÂI "

Let us look at the sign of the expression zm — zc

_ -1 - 12A\ + 2 v/l + 12A\ + 3 v/l - 8A\ + 16Af + 24A\C2 Zm ~Zc~ 36Â?

(4.4)

(4.5)

To prove that the expression (4.5) has strictly one sign for any values of A1 ^ 0.5, it is sufficient to consider the numerator of the fraction as a function E(A1 ,C2) of two variables:

E(AV C2) = -1 - 12A{ + 2^1 + V2A{ + ?>^1-8A{ + 16A\+2AA{C2. (4.6)

After replacing A1 with a nonnegat.ive parameter r = -\fl + 12A\ — 2, Eq. (4.6) takes the

form

E(r, C2) = -r2 - 2r + \J 18C2(1 + r)(3 + r) + r2(4 + r)2. (4.7)

If the inequality

r2 + 2r < \J 18C2(1 + r)(3 + r) + r2(4 + r)2 (4.8)

is true, inequality zTO > zc is also true. Since we are considering the case A1 ^ 0.5, the range of values of the parameter r belongs to the interval [0, to) and both sides of inequality (4.8) are positive. By squaring (4.8), we obtain

0 < 4r3 + 12r2 + 18C2r2 + 72C2r + 54C2, (4.9)

which is obviously true for all r ^ 0. Therefore, the expression (4.6) is positive for any A1 ^ 0.5. Since the numerator of the fraction (4.5) has a positive sign, we find that z3 > zm > zc, and the stationary point at sq = —1 is stable for any values of A1 > 0.

"I 0 I * (c)

Fig. 3. Phase portraits for three different flow regimes: (a) heteroclinic, (b) homoclinic, (c) transient

5. Separatrix reconnection bifurcation

The number of stationary points and the type of their stability in the system (3.1) do not depend on the control parameters. However, the structure of the phase space may differ due to the different type of connection of unstable points by separatrices: heteroclinic or homoclinic. In a heteroclinic connection, two chains of vortices in the north and south are separated by the area of the central jet, in which the flow is directed from east to west (see Fig. 3a). Physically, this mode corresponds to the situation where the phase velocity of the first Rossby wave, c1, (2.3), is less than the maximum flow velocity umax. If c1 > umax, then the separatrices are connected ho-moclinically, and in the region of the central jet the flow is directed from west to east (see Fig. 3b). When the control parameters change, a transition occurs between these two regimes through the bifurcation of reconnection of separatrices, when the northern and southern separatrices merge and the region of the central jet disappears (see Fig. 3c).

The condition under which the bifurcation occurs is determined by the equality of the streamfunction on both separatrices [3, 11]

^ (0, arctanh a/1 — z2) = ^ ^ j^-, — arctanh \/l — z.^j.

(5.1)

Using the expression (3.1) to determine the current function in (5.1) allows us to obtain an algebraic equation in the two parameters A1 and C2

-z2(Ai, C2)+ Ai(1 - z%(A1} C2))+ C2 arctanh z2(Ax, C2) = 0,

(5.2)

whose numerical solution is shown in Fig. 4.

0.8

0.6

¿i

0.4

0.2

0

Heteroclinic

0 0.1 0.2 0.3 0.4 0.5 0.6

<?2

Fig. 4. Bifurcation diagram

6. Conclusion

The present work is devoted to the study of the properties and structure of the phase space of the dynamically consistent model of a two-dimensional meandering jet obtained using the law of conservation of potential vorticity. Stationary points are analytically found in the phase space of

the advection equation and the type of their stability is determined. It is shown that, depending on the value of the control parameters, topologically different regimes are observed in the flow: with heteroclinic and homoclinic types of separatrix connection. A bifurcation curve is obtained numerically in the space of control parameters which separates the regions corresponding to different topological regimes of the jet flow (Fig. 4). The presented calculations are a standard technique for bifurcation analysis of low-dimensional Hamiltonian systems. The final list of topologically different flow modes is as follows:

1. If the values of the control parameters A1, C2 in the bifurcation diagram are chosen below the solution of Eq. (5.2), then there will be a heteroclinic flow regime. The central jet between the separatrices flows from east to west (see Fig. 3a).

2. When choosing the values of the control parameters in the bifurcation diagram above the solution of Eq. (5.2), a homoclinic flow regime will be observed in the system. The central jet between the separatrices flows from west to east (see Fig. 3b).

3. If the values of the parameters A1, C2 are solutions of the algebraic equation (5.2), then a transient flow regime will be observed. The area of the central jet disappears (see Fig. 3c).

Conflict of interest

The authors declare that they have no conflict of interest.

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