×

zbMATH — the first resource for mathematics

On the stability and bifurcation of the non-rotating Boussinesq equation with the Kolmogorov forcing at a low Péclet number. (English) Zbl 1444.76058
Summary: This study examines the stability and potential bifurcations of a stratified shear flow governed by the non-rotating incompressible Boussinesq equation at a low Péclet number. For the ratio of the vertical scale to the horizontal scale of a stratified flow \(a\in[\sqrt{3}/4,\sqrt{3}/2)\), it is shown that there exists a threshold for the Reynold number Re above which the steady stratified shear flow driven by the Kolmogorov forcing becomes linearly unstable. As a result, the Boussinesq equation exhibits a Hopf bifurcation. To further determine the type of the Hopf bifurcation, the model is reduced to a low-order system whose numerical analyses reveal that the Hopf bifurcation is supercritical. That is, a stable periodic solution emerges, which describes an oscillating thermal convection in a highly stratified shear flow arising in the atmosphere or interior of many stellar systems with low Péclet numbers.
MSC:
76E20 Stability and instability of geophysical and astrophysical flows
76E15 Absolute and convective instability and stability in hydrodynamic stability
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
35B32 Bifurcations in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Richardson, L. F., The supply of energy from and to atmospheric eddies, Proc R Soc Lond Ser A, 97, 354-373 (1920)
[2] Taylor, G. I., Effect of variation in density on the stability of superposed streams of fluid, Proc R Soc Lond Ser A, 132, 499-523 (1931) · Zbl 0002.42002
[3] Miles, J. W., On the stability of heterogeneous shear flows, J Fluid Mech, 10, 496-508 (1961) · Zbl 0101.43002
[4] Miles, J. W., On the stability of heterogeneous shear flows. II, J Fluid Mech, 16, 209-227 (1963) · Zbl 0123.22103
[5] Howard, L. N., Note on a paper of John W. Miles, J Fluid Mech, 10, 509-512 (1961) · Zbl 0104.20704
[6] Townsend, A. A., The effects of radiative transfer on turbulent flow of a stratified fluid, J Fluid Mech, 4, 361-375 (1958) · Zbl 0085.24003
[7] Zahn, J.-P., Rotational instabilities and stellar evolution, in stellar instability and evolution, JIAU Symp, 59, 185-194 (1974)
[8] Dudis, J. J., The stability of a thermally radiating stratified shear layer, including self-absorption, J Fluid Mech, 64, 65-83 (1974) · Zbl 0366.76038
[9] Spiegel, E. A.; Veronis, G., On the Boussinesq approximation for a compressible fluid, Astrophys J, 131, 442 (1960)
[10] Garaud, P.; Kumar, A.; Sridhar, J., The interaction between shear and fingering (thermohaline) convection, Astrophys J, 879, 60 (2019)
[11] Wang, Q.; Kieu, C., Dynamics of transverse cloud rolls in the boundary layer with the poiseuille shear flow, Phys Fluids, 31, 096601 (2019)
[12] Ibragimov, N. H.; Ibragimov, R. N., Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics, Commun Nonlinear Sci Numer Simul, 15, 8, 1989-2002 (2010) · Zbl 1222.86003
[13] Samanta, S.; Guha, A., Magnetohydrodynamic free convection flow above an isothermal horizontal plate, Commun Nonlinear Sci Numer Simul, 18, 12, 3407-3422 (2013) · Zbl 1344.76093
[14] Foias, C.; Manley, O.; Temam, R., Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal, 11, 8, 939-967 (1987) · Zbl 0646.76098
[15] Biswas, A.; Foias, C.; Larios, A., On the attractor for the semi-dissipative Boussinesq equations, Ann Inst H Poincaré Anal Non Linéaire, 34, 2, 381-405 (2017) · Zbl 1361.35138
[16] Pellew, A.; Southwell, R. V., On maintained convective motion in a fluid heated from below, Proc. Roy. Soc. London Ser. A, 176, 312-343 (1940) · JFM 66.1389.01
[17] Chandrasekhar, S., Hydrodynamic and hydromagnetic stability (2013), Courier Corporation · Zbl 0142.44103
[18] Kirchgässner, K., Bifurcation in nonlinear hydrodynamic stability, SIAM Rev, 17, 4, 652-683 (1975) · Zbl 0328.76035
[19] Ma, T.; Wang, S., Attractor bifurcation theory and its applications to Rayleigh-Bénard convection, Commun Pure Appl Anal, 2, 4, 591-599 (2003) · Zbl 1210.37056
[20] Rabinowitz, P. H., Existence and nonuniqueness of rectangular solutions of the Bénard problem, Arch Rational Mech Anal, 29, 32-57 (1968) · Zbl 0164.28704
[21] Qiu, H.; Du, Y.; Yao, Z., Blow-up criteria for 3D Boussinesq equations in the multiplier space, Commun Nonlinear Sci Numer Simul, 16, 4, 1820-1824 (2011) · Zbl 1221.35302
[22] Grady, K.; Gluhovsky, A., Exploring atmospheric convection with physically sound nonlinear low-order models, Commun Nonlinear Sci Numer Simul, 60, 128-136 (2018)
[23] Han, D.; Hernandez, M.; Wang, Q., Dynamic bifurcation and transition in the Rayleigh-Bénard convection with internal heating and varying gravity, Commun Math Sci, 17, 1, 175-192 (2019) · Zbl 1415.37097
[24] Spiegel, E. A., Thermal turbulence at very small Prandtl number, J Geophys Res, 67, 3063-3070 (1962)
[25] Thual, O., Zero-Prandtl-number convection,, J Fluid Mech, 240, 229-258 (1992) · Zbl 0775.76177
[26] Lignières, F., The small-Péclet-number approximation in stellar radiative zones, Astron Astrophys, 348, 933-939 (1999)
[27] Garaud, P.; Gallet, B.; Bischoff, T., The stability of stratified spatially periodic shear flows at low Péclet number, Phys Fluids, 27, 084104 (2015)
[28] Ma, T.; Wang, S., Phase transition dynamics (2014), Springer · Zbl 1285.82004
[29] S. Jiang, F.-F. J.; Ghil, M., Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model, J Phys Oceanogr, 25, 764-786 (1995)
[30] Meacham, S. P., Low-frequency variability in the wind-driven circulation, J Phys Oceanogr, 30, 269-293 (2000)
[31] Nadiga, B. T.; Luce, B. P., Global bifurcation of Shilnikov type in a double-gyre ocean model, J Phys Oceanogr, 31, 2669-2690 (2001)
[32] Simonnet, E.; Ghil, M.; Dijkstra, H., Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation, J Mar Res, 63, 931-956 (2005)
[33] Dijkstra, H.; Sengul, T.; Shen, J.; Wang, S., Dynamic transitions of quasi-geostrophic channel flow, SIAM J Appl Math, 75, 5, 2361-2378 (2015) · Zbl 1329.35232
[34] Gargano, F.; Ponetti, G.; Sammartino, M.; Sciacca., V., Route to chaos in the weakly stratified Kolmogorov flow, Phys Fluids, 31 (2019)
[35] Chen, Z.; Ghil, M.; Simonnet, E.; Wang, S., Hopf bifurcation in quasi-geostrophic channel flow, SIAM J Appl Math, 64, 343-368 (2003) · Zbl 1126.76327
[36] Chen, Z.-M.; Price, W. G., Supercritical regimes of liquid-metal fluid motions in electromagnetic fields: wall-bounded flows, R Soc Lond Proc Ser A, 458, 2027, 2735-2757 (2002) · Zbl 1015.76030
[37] Kieu, C.; Sengul, T.; Wang, Q.; Yan, D., On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents, Commun Nonlinear Sci Numer Simul, 65, 196-215 (2018)
[38] Wall, H. S., Analytic theory of continued fractions (2018), Courier Dover Publications · Zbl 1423.30002
[39] Meshalkin, L.; Sinai, I. G., Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid, J Appl Math Mech, 25, 6, 1700-1705 (1961) · Zbl 0108.39501
[40] Lu, C.; Mao, Y.; Wang, Q.; Yan, D., Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J Differ Equ, 267, 2560-2593 (2019) · Zbl 1415.35032
[41] Han, D.; Hernandez, M.; Wang, Q., On the instabilities and transitions of the western boundary current, Commun Comput Phys, 26, 35-56 (2019)
[42] Shen, J.; Tang, T.; Wang, L. L., Spectral methods: algorithms, analysis and applications, 41 (2011), Springer Science & Business Media
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.