Three-dimensional doubly diffusive convectons: instability and transition to complex dynamics.

*(English)*Zbl 1419.76214Summary: Three-dimensional doubly diffusive convection in a closed vertically extended container driven by competing horizontal temperature and concentration gradients is studied by a combination of direct numerical simulation and linear stability analysis. No-slip boundary conditions are imposed on all six container walls. The buoyancy number \(N\) is taken to be \(-1\) to ensure the presence of a conduction state. The primary instability is subcritical and generates two families of spatially localized steady states known as convectons. The convectons bifurcate directly from the conduction state and are organized in a pair of primary branches that snake within a well-defined range of Rayleigh numbers as the convectons grow in length. Secondary instabilities generating twist result in secondary snaking branches of twisted convectons. These destabilize the primary convectons and are responsible for the absence of stable steady states, localized or otherwise, in the subcritical regime. Thus all initial conditions in this regime collapse to the conduction state. As a result, once the Rayleigh number for the primary instability of the conduction state is exceeded, the system exhibits an abrupt transition to large-amplitude relaxation oscillations resembling bursts with no hysteresis. These numerical results are confirmed here by determining the stability properties of both convecton types as well as the domain-filling states. The number of unstable modes of both primary and secondary convectons of different lengths follows a pattern that allows the prediction of their stability properties based on their length alone. The instability of the convectons also results in a dramatic change in the dynamics of the system outside the snaking region that arises when the twist instability operates on a time scale faster than the time scale on which new rolls are nucleated. The results obtained are expected to be applicable in various pattern-forming systems exhibiting localized structures, including convection and shear flows.

##### MSC:

76E06 | Convection in hydrodynamic stability |

76R10 | Free convection |

76F06 | Transition to turbulence |

37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |

Full Text:
DOI

##### References:

[1] | Avila, M.; Mellibovsky, F.; Roland, N.; Hof, B., Streamwise-localized solutions at the onset of turbulence in pipe flow, Phys. Rev. Lett., 110, (2013) |

[2] | Avitabile, D.; Lloyd, D. J. B.; Burke, J.; Knobloch, E.; Sandstede, B., To snake or not to snake in the planar Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 9, 704-733, (2010) · Zbl 1200.37014 |

[3] | Batiste, O.; Knobloch, E.; Alonso, A.; Mercader, I., Spatially localized binary-fluid convection, J. Fluid Mech., 560, 149-158, (2006) · Zbl 1122.76029 |

[4] | Beaume, C., Adaptive Stokes preconditioning for steady incompressible flows, Commun. Comput. Phys., 22, 494-516, (2017) |

[5] | Beaume, C.; Bergeon, A.; Knobloch, E., Homoclinic snaking of localized states in doubly diffusive convection, Phys. Fluids, 23, (2011) |

[6] | Beaume, C.; Bergeon, A.; Knobloch, E., Convectons and secondary snaking in three-dimensional natural doubly diffusive convection, Phys. Fluids, 25, (2013) |

[7] | Beaume, C.; Knobloch, E.; Bergeon, A., Nonsnaking doubly diffusive convectons and the twist instability, Phys. Fluids, 25, (2013) |

[8] | Bergeon, A.; Knobloch, E., Natural doubly diffusive convection in three-dimensional enclosures, Phys. Fluids, 14, 3233-3250, (2002) · Zbl 1185.76049 |

[9] | Bergeon, A.; Knobloch, E., Periodic and localized states in natural doubly diffusive convection, Physica D, 237, 1139-1150, (2008) · Zbl 1138.76039 |

[10] | Bergeon, A.; Knobloch, E., Spatially localized states in natural doubly diffusive convection, Phys. Fluids, 20, (2008) · Zbl 1182.76055 |

[11] | Brand, E.; Gibson, J. F., A doubly localized equilibrium solution of plane Couette flow, J. Fluid Mech., 750, R3, (2014) |

[12] | Burke, J.; Dawes, J. H. P., Localized states in an extended Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 11, 261-284, (2012) · Zbl 1242.35047 |

[13] | Burke, J.; Knobloch, E., Localized states in the generalized Swift-Hohenberg equation, Phys. Rev. E, 73, (2006) · Zbl 1236.35144 |

[14] | Burke, J.; Knobloch, E., Homoclinic snaking: structure and stability, Chaos, 17, (2007) · Zbl 1163.37317 |

[15] | Burke, J.; Knobloch, E., Snakes and ladders: localized states in the Swift-Hohenberg equation, Phys. Lett. A, 360, 681-688, (2007) · Zbl 1236.35144 |

[16] | Dangelmayr, G.; Hettel, J.; Knobloch, E., Parity-breaking bifurcation in inhomogeneous systems, Nonlinearity, 74, 1093-1114, (1997) · Zbl 0995.34027 |

[17] | Duguet, Y.; Schlatter, P.; Henningson, D., Localized edge states in plane Couette flow, Phys. Fluids, 21, (2009) · Zbl 1183.76187 |

[18] | Ghorayeb, K.; Mojtabi, A., Double diffusive convection in a vertical regular cavity, Phys. Fluids, 9, 2339-2348, (1997) |

[19] | Gibson, J. F.; Brand, E., Spanwise-localized solutions of planar shear flows, J. Fluid Mech., 745, 25-61, (2014) |

[20] | Gibson, J. F.; Schneider, T. M., Homoclinic snaking in plane Couette flows: bending, skewing and finite-size effects, J. Fluid Mech., 794, 530-551, (2016) |

[21] | Hirschberg, P.; Knobloch, E., Mode interactions in large aspect ratio convection, J. Nonlinear Sci., 7, 537-556, (1997) · Zbl 0902.76035 |

[22] | Kao, H.-C.; Beaume, C.; Knobloch, E., Spatial localization in heterogeneous systems, Phys. Rev. E, 89, (2014) |

[23] | Khapko, T.; Kreilos, T.; Schlatter, P.; Duguet, Y.; Eckhardt, B.; Henningson, D., Localized edge states in the asymptotic suction boundary layer, J. Fluid Mech., 717, R6, (2013) · Zbl 1284.76106 |

[24] | Knobloch, E., Spatial localization in dissipative systems, Annu. Rev. Cond. Mat. Phys., 6, 325-359, (2015) |

[25] | Knobloch, E.; Hettel, J.; Dangelmayr, G., Parity breaking bifurcation in inhomogeneous systems, Phys. Rev. Lett., 74, 4839-4842, (1995) |

[26] | Lioubashevski, O.; Hamiel, Y.; Agnon, A.; Reches, Z.; Fineberg, J., Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension, Phys. Rev. Lett., 83, 3190-3193, (1999) |

[27] | Lloyd, D. J. B.; Gollwitzer, C.; Rehberg, I.; Richter, R., Homoclinic snaking near the surface instability of a polarizable fluid, J. Fluid Mech., 783, 283-305, (2015) · Zbl 1382.76291 |

[28] | Lo Jacono, D.; Bergeon, A.; Knobloch, E., Localized traveling pulses in natural doubly diffusive convection, Phys. Rev. Fluids, 2, (2017) |

[29] | Mellibovsky, F.; Meseguer, A., A mechanism for streamwise localisation of nonlinear waves in shear flows, J. Fluid Mech., 779, R1, (2015) · Zbl 1360.76095 |

[30] | Mercader, I.; Alonso, A.; Batiste, O., Spatiotemporal dynamics near the onset of convection for binary mixtures in cylindrical containers, Phys. Rev. E, 77, (2008) |

[31] | Mercader, I.; Batiste, O.; Alonso, A.; Knobloch, E., Localized pinning states in closed containers: Homoclinic snaking without bistability, Phys. Rev. E, 80, 025201(R), (2009) |

[32] | Mercader, I.; Batiste, O.; Alonso, A.; Knobloch, E., Convectons, anticonvectons and multiconvectons in binary fluid convection, J. Fluid Mech., 667, 586-606, (2011) · Zbl 1225.76107 |

[33] | Mercader, I.; Batiste, O.; Alonso, A.; Knobloch, E., Travelling convectons in binary fluid convection, J. Fluid Mech., 722, 240-266, (2013) · Zbl 1287.76105 |

[34] | Merkin, J. H.; Petrov, V.; Scott, S. K.; Showalter, K., Wave-induced chaos in a continuously fed unstirred reactor, J. Chem. Soc. Faraday Trans., 92, 2911-2918, (1996) |

[35] | Merkin, J. H.; Petrov, V.; Scott, S. K.; Showalter, K., Wave-induced chemical chaos, Phys. Rev. Lett., 76, 546-549, (1996) |

[36] | Nishiura, Y.; Ueyama, D., Spatio-temporal chaos for the Gray-Scott model, Physica D, 150, 137-162, (2001) · Zbl 0981.35022 |

[37] | Schneider, T. M.; Gibson, J. F.; Burke, J., Snakes and ladders: localized solutions of plane Couette flow, Phys. Rev. Lett., 104, (2010) |

[38] | Schneider, T. M.; Marinc, D.; Eckhardt, B., Localized edge states nucleate turbulence in extended plane Couette cells, J. Fluid Mech., 646, 441-451, (2010) · Zbl 1189.76258 |

[39] | Sezai, I.; Mohamad, A. A., Double diffusive convection in a cubic enclosure with opposing temperature and concentration gradients, Phys. Fluids, 12, 2210-2223, (2000) · Zbl 1184.76494 |

[40] | Thangam, S.; Zebib, A.; Chen, C. F., Double-diffusive convection in an inclined fluid layer, J. Fluid Mech., 116, 363-378, (1982) |

[41] | Watanabe, T.; Iima, M.; Nishiura, Y., Spontaneous formation of travelling localized structures and their asymptotic behaviour in binary fluid convection, J. Fluid Mech., 712, 219-243, (2012) · Zbl 1275.76095 |

[42] | Watanabe, T.; Iima, M.; Nishiura, Y., A skeleton of collision dynamics: Hierarchical network structure among even-symmetric steady pulses in binary fluid convection, SIAM J. Appl. Dyn. Syst., 15, 789-806, (2016) · Zbl 1375.76179 |

[43] | Woods, P. D.; Champneys, A. R., Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation, Physica D, 129, 147-170, (1999) · Zbl 0952.37009 |

[44] | Xin, S.; Le Quéré, P.; Tuckerman, L. S., Bifurcation analysis of double-diffusive convection with opposing horizontal thermal and solutal gradients, Phys. Fluids, 10, 850-858, (1998) |

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.