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Settling-driven large-scale instabilities in double-diffusive convection

Published online by Cambridge University Press:  24 August 2020

Raphael Ouillon Open the ORCID record for Raphael Ouillon [Opens in a new window] ,
Philip Edel ,
Pascale Garaud Open the ORCID record for Pascale Garaud [Opens in a new window]  and
Eckart Meiburg Open the ORCID record for Eckart Meiburg [Opens in a new window]
Raphael Ouillon*
Affiliation:
Mechanical Engineering, University of California, Santa Barbara, CA93106, USA
Philip Edel
Affiliation:
Mechanical Engineering, University of California, Santa Barbara, CA93106, USA ENSTA ParisTech, Université Paris-Saclay, Palaiseau, 91120, France
Pascale Garaud
Affiliation:
Applied Mathematics, Baskin School of Engineering, University of California, Santa Cruz, CA95064, USA
Eckart Meiburg
Affiliation:
Mechanical Engineering, University of California, Santa Barbara, CA93106, USA
*
†Email address for correspondence: [email protected]
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Abstract

When the density of a gravitationally stable fluid depends on a fast diffusing scalar and a slowly diffusing scalar of opposite contribution to the stability, ‘double diffusive’ instabilities may develop and drive convection. When the slow diffuser settles under gravity, as is for instance the case for small sediment particles in water, settling-driven double-diffusive instabilities can additionally occur. Such instabilities are relevant in a variety of naturally occurring settings, such as particle-laden river discharges, or underground inflows in lakes. Inspired by the dynamics of the more traditional thermohaline double-diffusive instabilities, we ask whether large-scale ‘mean-field’ instabilities can develop as a result of sedimentary double-diffusive convection. We first apply the mean-field instability theory of Traxler et al. (J. Fluid Mech., vol. 677, 2011, pp. 530–553) to high-Prandtl-number fluids, and find that these are unstable to Radko's layering instability, yet collectively stable. We then extend the theory of Traxler et al. (2011) to include settling and study its impact on the development of the collective instability. We find that two distinct regimes exist. At low settling velocities, the double-diffusive turbulence in the fingering regime is relatively unaffected by settling and remains stable to the classical collective instability. It is, however, unstable to a new instability in which large-scale gravity waves are excited by the phase shift between the salinity and particle concentration fields. At higher settling velocities, the double-diffusive turbulence is substantially affected by settling, and becomes unstable to the classic collective instability. Our findings, validated by direct numerical simulations, reveal new opportunities to observe settling-driven layering in laboratory and field experiments.

JFM classification

Convection: Double diffusive convection Geophysical and Geological Flows: Stratified flows Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 901 , 25 October 2020 , A12
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Alsinan, A., Meiburg, E. & Garaud, P. 2017 A settling-driven instability in two-component, stably stratified fluids. J. Fluid Mech. 816, 243–267.CrossRefGoogle Scholar
Baines, P. G. & Gill, A. E. 1969 On thermohaline convection with linear gradients. J. Fluid Mech. 37 (2), 289–306.CrossRefGoogle Scholar
Burns, P. & Meiburg, E. 2012 Sediment-laden fresh water above salt water: linear stability analysis. J. Fluid Mech. 691, 279–314.CrossRefGoogle Scholar
Burns, P. & Meiburg, E. 2015 Sediment-laden fresh water above salt water: nonlinear simulations. J. Fluid Mech. 762, 156–195.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2007 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Carazzo, G. & Jellinek, M. A. 2013 Particle sedimentation and diffusive convection in volcanic ash-clouds. J. Geophys. Res. 118 (4), 1420–1437.CrossRefGoogle Scholar
Davarpanah Jazi, S. & Wells, M. G. 2016 Enhanced sedimentation beneath particle-laden flows in lakes and the ocean due to double-diffusive convection. Geophys. Res. Lett. 43 (20), 10883–10890.CrossRefGoogle Scholar
Davis, R. H. 1996 Hydrodynamic diffusion of suspended particles: a symposium. J. Fluid Mech. 310, 325–335.CrossRefGoogle Scholar
Ferry, J. & Balachandar, S. 2001 A fast Eulerian method for disperse two-phase flow. Intl J. Multiphase Flow 27 (7), 1199–1226.CrossRefGoogle Scholar
Garaud, P. 2018 Double-diffusive convection at low Prandtl Number. Annu. Rev. Fluid Mech. 50 (1), 275–298.CrossRefGoogle Scholar
Garaud, P., Medrano, M., Brown, J. M., Mankovich, C. & Moore, K. 2015 Excitation of gravity waves by fingering convection, and the formation of compositional staircases in stellar interiors. Astrophys. J. 808 (1), 1–14.CrossRefGoogle Scholar
Green, T. 1987 The importance of double diffusion to the settling of suspended material. Sedimentology 34 (2), 319–331.CrossRefGoogle Scholar
Holyer, J. Y. 1981 On the collective instability of salt fingers. J. Fluid Mech. 110, 195–207.CrossRefGoogle Scholar
Houk, D. & Green, T. 1973 Descent rates of suspension fingers. Deep-Sea Res. Oceanogr. Abstr. 20 (8), 757–761.CrossRefGoogle Scholar
Krishnamurti, R. 2003 Double-diffusive transport in laboratory thermohaline staircases. J. Fluid Mech. 483, 287–314.CrossRefGoogle Scholar
Kunze, E. 2003 A review of oceanic salt-fingering theory. Prog. Oceanogr. 56 (3–4), 399–417.CrossRefGoogle Scholar
Linden, P. 1973 On the structure of salt fingers. Deep-Sea Res. Oceanogr. Abstr. 20 (4), 325–340.CrossRefGoogle Scholar
Merryfield, W. J. 2000 Origin of thermohaline staircases. J. Phys. Oceanogr. 30 (5), 1046–1068.2.0.CO;2>CrossRefGoogle Scholar
Peyret, R. 2002 Spectral methods for incompressible viscous flow. Applied Mathematical Sciences, vol. 148. Springer.CrossRefGoogle Scholar
Poisson, A. & Papaud, A. 1983 Diffusion coefficients of major ions in seawater. Mar. Chem. 13 (4), 265–280.CrossRefGoogle Scholar
Radko, T. 2003 A mechanism for layer formation in a double-diffusive fluid. J. Fluid Mech. 497, 365–380.CrossRefGoogle Scholar
Radko, T. 2013 Double-Diffusive Convection. Cambridge University Press.CrossRefGoogle Scholar
Radko, T. & Smith, D. P. 2012 Equilibrium transport in double-diffusive convection. J. Fluid Mech. 692, 5–27.CrossRefGoogle Scholar
Rani, S. L. & Balachandar, S. 2003 Evaluation of the equilibrium Eulerian approach for the evolution of particle concentration in isotropic turbulence. Intl J. Multiphase Flow 29 (12), 1793–1816.CrossRefGoogle Scholar
Reali, J. F., Garaud, P., Alsinan, A. & Meiburg, E. 2017 Layer formation in sedimentary fingering convection. J. Fluid Mech. 816, 268–305.CrossRefGoogle Scholar
Rudnick, D. L. 1999 Compensation of horizontal temperature and salinity gradients in the ocean mixed layer. Science 283 (5401), 526–529.CrossRefGoogle ScholarPubMed
Schmitt, R. W. 1979 The growth rate of super-critical salt fingers. Deep-Sea Res. I 26 (1), 23–40.CrossRefGoogle Scholar
Schmitt, R. W. 1994 Double diffusion in oceanography. Annu. Rev. Fluid Mech. 26 (1), 255–285.CrossRefGoogle Scholar
Schmitt, R. W., Perkins, H., Boyd, J. D. & Stalcup, M. C. 1987 C-SALT: an investigation of the thermohaline staircase in the western tropical North Atlantic. Deep-Sea Res. I 34 (10), 1655–1665.CrossRefGoogle Scholar
Shen, C. Y. & Schmitt, R. W. 2013 The salt finger wavenumber spectrum. In Double-Diffusive Convection (eds A. Brandt & H. Fernando). doi:10.1029/GM094p0305.CrossRefGoogle Scholar
Stellmach, S. & Hansen, U. 2008 An efficient spectral method for the simulation of dynamos in Cartesian geometry and its implementation on massively parallel computers. Geochem. Geophys. Geosyst. 9 (5), 1–11.CrossRefGoogle Scholar
Stellmach, S., Traxler, A., Garaud, P., Brummell, N. & Radko, T. 2011 Dynamics of fingering convection. Part 2. The formation of thermohaline staircases. J. Fluid Mech. 677, 554–571.CrossRefGoogle Scholar
Stern, M. E. 1960 The “salt-fountain” and thermohaline convection. Tellus 12 (2), 172–175.CrossRefGoogle Scholar
Stern, M. E. 1969 Collective instability of salt fingers. J. Fluid Mech. 35 (02), 209–218.CrossRefGoogle Scholar
Stern, M. E., Radko, T. & Simeonov, J. 2001 Salt fingers in an unbounded thermocline. J. Mar. Res. 59 (3), 355–390.CrossRefGoogle Scholar
Sánchez, X. & Roget, E. 2007 Microstructure measurements and heat flux calculations of a triple-diffusive process in a lake within the diffusive layer convection regime. J. Geophys. Res. 112 (C2), C02012.CrossRefGoogle Scholar
Tait, R. I. & Howe, M. R. 1968 Some observations of thermo-haline stratification in the deep ocean. Deep-Sea Res. Oceanogr. Abstr. 15 (3), 275–280.CrossRefGoogle Scholar
Traxler, A., Stellmach, S., Garaud, P., Radko, T. & Brummell, N. 2011 Dynamics of fingering convection. Part 1. Small-scale fluxes and large-scale instabilities. J. Fluid Mech. 677, 530–553.CrossRefGoogle Scholar
Turner, J. S. 1967 Salt fingers across a density interface. Deep-Sea Res. Oceanogr. Abstr. 14 (5), 599–611.CrossRefGoogle Scholar
Turner, J. S. 1974 Double-diffusive phenomena. Annu. Rev. Fluid Mech. 6 (1), 37–54.CrossRefGoogle Scholar
Walsh, D. & Ruddick, B. 1995 Double-diffusive interleaving: the influence of nonconstant fiffusivities. J. Phys. Oceanogr. 25 (3), 348–358.2.0.CO;2>CrossRefGoogle Scholar
Whitfield, D. W. A., Holloway, G. & Holyer, J. Y. 1989 Spectral transform simulations of finite amplitude double-diffusive instabilities in two dimensions. J. Mar. Res. 47 (2), 241–265.CrossRefGoogle Scholar
You, Y. 2002 A global ocean climatological atlas of the Turner angle: implications for double-diffusion and water-mass structure. Deep-Sea Res. I 49 (11), 2075–2093.CrossRefGoogle Scholar
Yu, X., Hsu, T. & Balachandar, S. 2013 Convective instability in sedimentation: linear stability analysis. J. Geophys. Res. 118 (1), 256–272.CrossRefGoogle Scholar
Yu, X., Hsu, T. & Balachandar, S. 2014 Convective instability in sedimentation: 3-D numerical study. J. Geophys. Res. 119 (11), 8141–8161.CrossRefGoogle Scholar