Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-29T08:21:04.588Z Has data issue: false hasContentIssue false
  • English
  • Français

Jet formation in salt-finger convection: a modified Rayleigh–Bénard problem

Published online by Cambridge University Press:  02 November 2018

Jin-Han Xie Open the ORCID record for Jin-Han Xie [Opens in a new window] ,
Keith Julien  and
Edgar Knobloch
Jin-Han Xie*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Keith Julien
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Edgar Knobloch
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA
*
†Email address for correspondence: jhxie@cims.nyu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Large-scale coherent structures such as jets in Rayleigh–Bénard convection and related systems are receiving increasing attention. This paper studies, both numerically and theoretically, the process of jet formation in two-dimensional salt-finger convection. The approach utilizes an asymptotically derived system of equations referred to as the modified Rayleigh–Bénard convection (MRBC) model, valid in the geophysically and astrophysically relevant limit in which the solute diffuses much more slowly than heat. In these equations, convection is driven by a destabilizing salinity gradient while the effects of the stabilizing temperature gradient manifest themselves as an additional anisotropic dissipation acting on large scales. The MRBC system is specified by two external parameters: the Schmidt number $\mathit{Sc}$ (ratio of viscosity to solutal diffusivity) and the Rayleigh ratio $\mathit{Ra}$ (ratio between the Rayleigh numbers of the destabilizing solutal stratification and the stabilizing thermal stratification). Two distinct $\mathit{Ra}$ regimes are explored for fixed $\mathit{Sc}=1$. In all cases studied the system develops a horizontal jet structure that is maintained self-consistently by turbulent fluctuations, but coarsens over time. For intermediate Rayleigh ratios (e.g. $\mathit{Ra}=6$), the MRBC model captures the relaxation oscillations superposed on the jet structure observed at similar parameter values in direct numerical simulations of the primitive equations. For smaller Rayleigh ratios (e.g. $\mathit{Ra}=2$), a regime for which direct numerical simulation of the primitive equations is difficult because of the presence of fast gravity waves, the MRBC model reveals the existence of statistically steady jets whose properties are studied in detail. Three hierarchical models, the MRBC and further reductions in the form of quasilinear and single-mode approximations, are used to confirm that jets form and are sustained as a result of the interaction between fluctuations (salt fingers) and large-scale horizontally averaged horizontal flows (jets). Even though the small-scale structures exhibited by the three models exhibit clear differences, all three produce the same power-law spectrum of the mean fields at large vertical scales: in all, the spectrum of the mean streamfunction scales as $m^{-3}$ and the mean salinity field scales as $m^{-1}$, with $m$ the vertical wavenumber. A theoretical explanation of these observations based on the dominant balances in the mean and fluctuation equations is provided. As a consequence, the jets have a zigzag profile, a conclusion that is consistent with numerical simulations. Based on numerical observations, a three-component phenomenological model consisting of a linearly growing mode, a linearly damped mode and a mean mode is proposed to explain the observed transition from statistically steady jet structure to jets with superposed oscillations that takes place with increasing Rayleigh ratio.

JFM classification

Convection: Convection Convection: Double diffusive convection Wakes/Jets: Jets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 858 , 10 January 2019 , pp. 228 - 263
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balmforth, N. J., Llewellyn Smith, S. G. & Young, W. R. 1998 Dynamics of interfaces and layers in a stratified turbulent fluid. J. Fluid Mech. 355, 329–358.Google Scholar
Beaume, C., Chini, G. P., Julien, K. & Knobloch, E. 2015 Reduced description of exact coherent states in parallel shear flows. Phys. Rev. E 91, 043010.Google Scholar
Bouchet, F. & Simonnet, E. 2009 Random changes of flow topology in two-dimensional and geophysical turbulence. Phys. Rev. Lett. 102, 094504.Google Scholar
Bouchet, F. & Venaille, A. 2012 Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515, 227–295.Google Scholar
Brown, J. M., Garaud, P. & Stellmach, S. 2013 Chemical transport and spontaneous layer formation in fingering convection in astrophysics. Astrophys. J. 768, 34.Google Scholar
Brummell, N. H. & Hart, J. E. 1993 High Rayleigh number 𝛽-convection. Geophys. Astrophys. Fluid Dyn. 68, 85–114.Google Scholar
Busse, F. H. 1983 A model of mean zonal flows in the major planets. Geophys. Astrophys. Fluid Dyn. 23, 153–174.Google Scholar
Childress, S. 2000 Eulerian mean flow from an instability of convective plumes. Chaos 10, 28–38.Google Scholar
Chong, K. L., Yang, Y., Huang, S.-D., Zhong, J.-Q., Stevens, R. J. A. M., Verzicco, R., Lohse, D. & Xia, K.-Q. 2017 Confined Rayleigh–Bénard, rotating Rayleigh–Bénard, and double diffusive convection: a unifying view on turbulent transport enhancement through coherent structure manipulation. Phys. Rev. Lett. 119, 064501.Google Scholar
Denissenkov, P. A. 2010 Numerical simulations of thermohaline convection: implications for extra-mixing in low-mass RGB stars. Astrophys. J. 723, 563–579.Google Scholar
Falkovich, G. 2016 Interaction between mean flow and turbulence in two dimensions. Proc. R. Soc. Lond. A 472, 20160287.Google Scholar
Farrell, B. F. & Ioannou, P. J. 2003 Structural stability of turbulent jets. J. Atmos. Sci. 50, 2101–2118.Google Scholar
Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64, 3652–3665.Google Scholar
Fauve, S., Herault, J., Michel, G. & Pétrélis, F. 2017 Instabilities on a turbulent background. J. Stat. Mech.: Theory Exp. 6, 064001.Google Scholar
Fernandes, A. M. & Krishnamurti, R. 2010 Salt finger fluxes in a laminar shear flow. J. Fluid Mech. 658, 148–165.Google Scholar
Fitzgerald, J. G. & Farrell, B. F. 2014 Mechanisms of mean flow formation and suppression in two-dimensional Rayleigh–Bénard convection. Phys. Fluids 26, 054104.Google Scholar
Francois, N., Xia, H., Punzmann, H. & Shats, M. 2013 Inverse energy cascade and emergence of large coherent vortices in turbulence driven by Faraday waves. Phys. Rev. Lett. 110, 194501.Google Scholar
Frishman, A. 2017 The culmination of an inverse cascade: mean flow and fluctuations. Phys. Fluids 29, 125102.Google Scholar
Frishman, A., Laurie, J. & Falkovich, G. 2017 Jets or vortices: What flows are generated by an inverse turbulent cascade? Phys. Rev. Fluids 2, 032602(R).Google Scholar
Garaud, P. 2018 Double-diffusive convection at low Prandtl number. Annu. Rev. Fluid Mech. 50, 275–298.Google Scholar
Garaud, P. & Brummell, N. 2015 2D or not 2D: the effect of dimensionality on the dynamics of fingering convection at low Prandtl number. Astrophys. J. 815, 42.Google Scholar
Goluskin, D., Johnston, H., Flierl, G. R. & Spiegel, E. A. 2014 Convectively driven shear and decreased heat flux. J. Fluid Mech. 795, 360–385.Google Scholar
Guervilly, C. & Hughes, D. W. 2017 Jets and large-scale vortices in rotating Rayleigh–Bénard convection. Phys. Rev. Fluids 2, 113503.Google Scholar
von Hardenberg, J., Goluskin, D., Provenzale, A. & Spiegel, E. A. 2015 Generation of large-scale winds in horizontally anisotropic convection. Phys. Rev. Lett. 115, 134501.Google Scholar
Hermiz, K. B., Guzdar, P. N. & Finn, J. M. 1995 Improved low-order model for shear flow driven by Rayleigh–Bénard convection. Phys. Rev. E 51, 325–331.Google Scholar
Holyer, J. Y. 1984 The stability of long, steady, two-dimensional salt fingers. J. Fluid Mech. 147, 169–185.Google Scholar
Horton, W., Hu, G. & Laval, G. 1996 Turbulent transport in mixed states of convective cells and sheared flows. Phys. Plasmas 3, 2912–2923.Google Scholar
Howard, L. N. & Krishnamurti, R. 1986 Large-scale flow in turbulent convection: a mathematical model. J. Fluid Mech. 170, 385–410.Google Scholar
Hughes, D. W. & Proctor, M. R. E. 1990 A low-order model of the shear instability of convection: chaos and the effect of noise. Nonlinearity 3, 127–153.Google Scholar
Julien, K., Knobloch, E. & Plumley, M. 2018 Impact of domain anisotropy on the inverse cascade in geostrophic turbulent convection. J. Fluid Mech. 834, R4.Google Scholar
Krishnamurti, R. & Howard, L. N. 1981 Large-scale flow generation in turbulent convection. Proc. Natl Acad. Sci. USA 78, 1981–1985.Google Scholar
Laurie, J., Boffetta, G., Falkovich, G., Kolokolov, I. & Lebedev, V. 2014 Universal profile of the vortex condensate in two-dimensional turbulence. Phys. Rev. Lett. 113, 254530.Google Scholar
Linden, P. F. 1974 Salt fingers in a steady shear flow. Geophys. Fluid Dyn. 6, 1–27.Google Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 185–195.Google Scholar
Marston, J. B., Conover, E. & Schneider, T. 2008 Statistics of an unstable barotropic jet from a cumulant expansion. J. Atmos. Sci. 65, 1955–1966.Google Scholar
Matthews, P. C., Proctor, M. R. E., Rucklidge, A. M. & Weiss, N. O. 1993 Pulsating waves in nonlinear magnetoconvection. Phys. Lett. A 183, 69–75.Google Scholar
Matthews, P. C., Rucklidge, A. M., Weiss, N. O. & Proctor, M. R. E. 1996 The three-dimensional development of the shearing instability. Phys. Fluids 8, 1350–1352.Google Scholar
Merryfield, W., Holloway, G. & Gargett, A. 1999 A global ocean model with double-diffusive mixing. J. Phys. Oceanogr. 29, 1124–1142.Google Scholar
Mujica, N. & Lathrop, D. P. 2015 Hysteretic gravity-wave bifurcation in a highly turbulent swirling flow. J. Fluid Mech. 551, 49–62.Google Scholar
O’Gorman, P. A. & Schneider, T. 2007 Recovery of atmospheric flow statistics in a general circulation model without nonlinear eddy–eddy interactions. Geophys. Res. Lett. 34, L22801.Google Scholar
Paparella, F. & von Hardenberg, J. 2012 Clustering of salt fingers in double-diffusive convection leads to staircaselike stratification. Phys. Rev. Lett. 109, 014502.Google Scholar
Paparella, F. & Spiegel, E. A. 1999 Sheared salt fingers: instability in a truncated system. Phys. Fluids 11, 1161–1168.Google Scholar
Phillips, O. M. 1958 The equilibrium range in the spectrum of wind-generated waves. J. Fluid Mech. 4, 426–434.Google Scholar
Prat, V., Lignières, F. & Lagarde, N. 2015 Numerical simulations of zero-Prandtl-number thermohaline convection. In SF2A-2015: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, pp. 419–422. Société Française d’Astronomie et d’Astrophysique.Google Scholar
Radko, T. 2003 A mechanism for layer formation in a double-diffusive fluid. J. Fluid Mech. 497, 365–380.Google Scholar
Radko, T. 2008 The double-diffusive modon. J. Fluid Mech. 609, 59–85.Google Scholar
Radko, T. 2010 Equilibration of weakly nonlinear salt fingers. J. Fluid Mech. 645, 121–143.Google Scholar
Radko, T. 2013 Double-diffusive Convection. Cambridge University Press.Google Scholar
Radko, T., Ball, J., Colosi, J. & Flanagan, J. 2015 Double-diffusive convection in a stochastic shear. J. Phys. Oceanogr. 45, 3155–3167.Google Scholar
Radko, T. & Smith, D. P. 2012 Equilibrium transport in double-diffusive convection. J. Fluid Mech. 692, 5–27.Google Scholar
Radko, T. & Stern, M. E. 1999 Salt fingers in three dimensions. J. Mar. Res. 57, 471–502.Google Scholar
Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443.Google Scholar
Rucklidge, A. M. & Matthews, P. C. 1996 Analysis of the shearing instability in nonlinear convection and magnetoconvection. Nonlinearity 9, 311–351.Google Scholar
Schmitt, R. W. 1983 The characteristics of salt fingers in a variety of fluid systems, including stellar interiors, liquid metals, oceans, and magmas. Phys. Fluids 26, 2373–2377.Google Scholar
Schmitt, R. W. 1994 Double diffusion in oceanography. Annu. Rev. Fluid Mech. 26, 255–285.Google Scholar
Shen, C. Y. 1995 Equilibrium salt-fingering convection. Phys. Fluids 7, 704–717.Google Scholar
Smith, L. M. & Yakhot, V. 1994 Finite-size effects in forced two-dimensional turbulence. J. Fluid Mech. 274, 115–138.Google Scholar
Smyth, W. & Kimura, S. 2007 Instability and diapycnal momentum transport in a double-diffusive, stratified shear layer. J. Phys. Oceanogr. 37, 1551–1565.Google Scholar
Srinivasan, K. & Young, W. R. 2012 Zonostrophic instability. J. Atmos. Sci. 69, 1633–1656.Google 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.Google Scholar
Stern, M. E. 1960 The salt-fountain and thermohaline convection. Tellus 12, 172–175.Google Scholar
Stern, M. E. 1969 Collective instability of salt fingers. J. Fluid Mech. 35, 209–218.Google Scholar
Stern, M. E. & Radko, T. 1998 The salt finger amplitude in unbounded T–S gradient layers. J. Mar. Res. 56, 157–196.Google Scholar
Traxler, A., Garaud, P. & Stellmach, S. 2011a Numerically determined transport laws for fingering (‘thermohaline’) convection in astrophysics. Astrophys. J. 728, L29.Google Scholar
Traxler, A., Stellmach, S., Garaud, P., Radko, T. & Brummell, N. 2011b Dynamics of fingering convection. Part 1. Small-scale fluxes and large-scale instabilities. J. Fluid Mech. 677, 530–553.Google Scholar
Turner, J. S. 1974 Double diffusive phenomena. Annu. Rev. Fluid Mech. 6, 37–54.Google Scholar
Xia, H. & Francois, N. 2017 Two-dimensional turbulence in three-dimensional flows. Phys. Fluids 29, 111107.Google Scholar
Xie, J.-H., Knobloch, E., Miquel, B. & Julien, K. 2017 A reduced model for salt-finger convection in the small diffusivity ratio limit. Fluids 2 (1), 6.Google Scholar