## On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents.(English)Zbl 1456.76055

Summary: This study examines the instability and dynamical transitions of the two-layer western boundary currents represented by the Munk profile in the upper layer and a motionless bottom layer in a closed rectangular domain. First, a bound on the intensity of the Munk profile below which the western boundary currents are locally nonlinearly stable is provided. Second, by reducing the infinite dimensional system to a finite dimensional one via the center manifold reduction, non-dimensional transition numbers are derived, which determine the types of dynamical transitions both from a pair of simple complex eigenvalues as well as from a double pair of complex conjugate eigenvalues as the Reynolds number crosses a critical threshold. We show by careful numerical estimations of the transition numbers that the transitions in both cases are continuous at the critical Reynolds number. After the transition from a pair of simple complex eigenvalue, the western boundary layer currents turn into a periodic circulation, whereas a quasi-periodic or possibly a chaotic circulation emerges after the transition from a pair of double complex eigenvalues. Finally, a comparison between the transitions exhibited in one-layer and two-layer models is provided, which demonstrates the fundamental differences between the two models.

### MSC:

 76E20 Stability and instability of geophysical and astrophysical flows 86A05 Hydrology, hydrography, oceanography
Full Text:

### References:

 [1] Charney, J. G., The gulf stream as an inertial boundary layer, Pro Natl Acad Sci, 41, 731-740 (1955) [2] Stommel, H., The Gulf stream: A Physical and dynamical description (1965), California library reprint series. University of California Press [3] Stramma, L.; Lutjeharms, J. R.E., The flow field of the subtropical gyre of the south indian ocean, J Geophy Res: Oceans, 102, C3, 5513-5530 (1997) [4] Bryan, K., A numerical investigation of nonlinear model of wind-driven ocean, J Atmos Sci, 20, 594-606 (1963) [5] Kamenkovich, V. M.; Beloyserkovskii, S. O.; Panteleev, M. C., On the problem of numerical modeling a barotropic current generated by a large scale wind field, Izv POLYMODE, 15, 3-23 (1985) [6] Panteleev, M. C., The influence of friction on the character of the barotropic wind driven circulation, Izv POLYMODE, 15, 34-39 (1985) [7] Ierley, G. R., On the onset of inertial recirculation in barotropic general circulation models, J Phys Oceanogr, 17, 12, 2366-2374 (1987) [8] Berloff, P.; Meacham, S., On the stability of the wind-driven circulation, J Mar Res, 56, 5, 937-993 (1998) [9] Ghil, M., The wind-driven ocean circulation: applying dynamical systems theory to a climate problem, Discrete Contin Dyn Syst-A, 37, 1, 189-228 (2017) [10] Pedlosky, J., Geophysical fluid dynamics (1987), Springer [11] Stommel, H., The westward intensification of wind-driven ocean currents, Eos (Washington DC), 29, 202-206 (1948) [12] Munk, W. H., On the wind-driven ocean circulation, J Meteor, 7, 2, 79-93 (1950) [13] Veronis, G., Wind-driven ocean circulation: part 1. linear theory and perturbation analysis, Deep-Sea Res, 13, 1, 17-29 (1966) [14] Veronis, G., Wind-driven ocean circulation: part 2. numerical solutions of the non-linear problem, Deep-Sea Res, 13, 1, 31-55 (1966) [15] Simonnet, E.; Ghil, M.; Ide, K.; Temam, R.; Wang, S., Low-frequency variability in shallow-water models of the wind-driven ocean circulation. part i: steady-state solution, J Phys Oceanogr, 33, 712-728 (2003) [16] Simonnet, E.; Ghil, M.; Ide, K.; Temam, R.; Wang, S., Low-frequency variability in shallow-water models of the wind-driven ocean circulation. part ii: time-dependent solutions, J Phys Oceanogr, 33, 729-752 (2003) [17] Ierley, G. R.; Young, W. R., Viscous instabilities in the western boundary layer, J Phys Oceanogr, 21, 9, 1323-1332 (1991) [18] Ma, T.; Wang, S., Phase transition dynamics (2014), Springer [19] Dijkstra, H.; Sengul, T.; Shen, J.; Wang, S., Dynamic transitions of quasi-geostrophic channel flow, SIAM J Appl Math, 75, 5, 2361-2378 (2015) [20] Ma, T.; Wang, S., Rayleigh-bénard convection: dynamics and structure in the physical space, Commun Math Sci, 5, 3, 553-574 (2007) [21] Sengul, T.; Wang, S., Pattern formation in rayleigh-bénard convection, Commun Math Sci, 11, 1, 315-343 (2013) [22] Liu, H.; Sengul, T.; Wang, S., Dynamic transitions for quasilinear systems and cahn-hilliard equation with onsager mobility, J Math Phys, 53, 2, 023518,31 (2012) [23] Liu, H.; Sengul, T.; Wang, S.; Zhang, P., Dynamic transitions and pattern formations for a Cahn-Hilliard model with long-range repulsive interactions, Commun Math Sci, 13, 5, 1289-1315 (2015) [24] Ma, T.; Wang, S., Boundary-layer and interior separations in the taylor-couette-poiseuille flow, J Math Phys, 50, 3, 033101,29 (2009) [25] S. Attili, B., Numerical treatment of singularly perturbed two point boundary value problems exhibiting boundary layers, Commun Nonlinear Sci Numer Simul, 16, 9, 3504-3511 (2011) [26] Wang, Q., Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders, Discrete Contin Dyn Syst-B, 19, 2, 543-563 (2014) [27] Li, L.; Hernandez, M.; Ong, K. W., Stochastic attractor bifurcation for the two-dimensional swift-hohenberg equation, Math Methods Appl Sci, 41, 5, 2105-2118 (2018) [28] Hernández, M.; Ong, K. W., Stochastic Swift-Hohenberg equation with degenerate linear multiplicative noise, J Math Fluid Mech, 1-20 (2018) [29] Ozer, S.; Sengul, T., Stability and transitions of the second grade poiseuille flow, Phys D, 331, 71-81 (2016) [30] Majda, A.; Wang, X., Nonlinear dynamics and statistical theories for basic geophysical flows (2006), Cambridge University Press [31] Henry, D., Geometric theory of semilinear parabolic equations, 840 (1981), Springer [32] Kuznetsov, Y. A., Elements of applied bifurcation theory, 112 (2013), Springer [33] Shen, J.; Tang, T.; Wang, L.-L., Spectral methods: algorithms, analysis and applications, 41 (2011), Springer [34] Sengul, T.; Shen, J.; Wang, S., Pattern formations of 2d rayleigh-bénard convection with no-slip boundary conditions for the velocity at the critical length scales, Math Methods Appl Sci, 38, 17, 3792-3806 (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.