Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T10:46:07.167Z Has data issue: false hasContentIssue false
  • English
  • Français

Double-diffusive instabilities at a horizontal boundary after the sudden onset of heating

Published online by Cambridge University Press:  16 November 2018

Oliver S. Kerr Open the ORCID record for Oliver S. Kerr [Opens in a new window]
Oliver S. Kerr*
Affiliation:
Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, UK
*
†Email address for correspondence: o.s.kerr@city.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

When a deep body of fluid with a stable salinity gradient is heated from below at a horizontal boundary a destabilizing temperature gradient develops and can lead to instabilities. We will focus on two variants of this problem: the sudden increase in the boundary temperature at the initial time and the sudden turning on of a constant heat flux. These generate time-dependent temperature profiles. We look at the growing phase of the linear instabilities as an initial value problem where the initial time for the instabilities is a parameter to be determined. We determine numerically the optimal initial conditions and the optimal starting time for the instabilities to ensure that the maximum growth occurs at some given later time. The method that is used is an extension of the method developed by Kerr & Gumm (J. Fluid Mech., vol. 825, 2017, pp. 1002–1034) in their investigation of the stability of developing temperature boundary layers at horizontal and vertical boundaries. This requires the use of an appropriate measure of the amplitude of the disturbances which is identified. The effectiveness of this approach is verified by looking at the classic problem of double-diffusive convection in a horizontal layer, where we look at both the salt-finger regime and the diffusive regime. We show that this approach is an effective way of investigating instabilities where the background gradients time dependent. For the problem of heating a salinity gradient from below, as the heat diffuses into the fluid the effective thermal Rayleigh number based on the instantaneous diffusion length scale grows. For the case of a sudden increase in the temperature by a fixed amount the effective thermal Rayleigh number is proportional to $t^{3/2}$, and for a constant heat flux it is proportional to $t^{2}$, where $t$ is the time since the onset of heating. However, the effective salt Rayleigh number also grows as $t^{2}$. We will show that for the constant temperature case the thermal Rayleigh number initially dominates and the instabilities undergo a phase where the convection is essentially thermal, and the onset is essentially instantaneous. As the salt Rayleigh number becomes more significant the instability undergoes a transition to oscillatory double-diffusive convection. For the constant heat flux the ratio of the thermal and salt Rayleigh numbers is constant, and the instabilities are always double diffusive in their nature. These instabilities initially decay. Hence, to achieve the largest growth at some given fixed time, there is an optimal time after the onset of heating for the instabilities to be initiated. These instabilities are essentially double diffusive throughout their growth.

JFM classification

Convection: Convection Convection: Double diffusive convection Instability: Instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 859 , 25 January 2019 , pp. 126 - 159
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

Baines, P. G. & Gill, A. E. 1969 On thermohaline convection with linear gradients. J. Fluid Mech. 37, 289–306.Google Scholar
Bretherton, C. S. 1981 Double diffusion in a long box. In Woods Hole Geophysical Fluid Dynamics Course Lectures WHOI-81-102 (ed. Mellor, F. K.), pp. 201–234. Woods Hole Oceanographic Institution.Google Scholar
Chan, C. L., Chen, W.-Y. & Chen, C. F. 2002 Secondary motion in convection layers generated by lateral heating of a solute gradient (with an Appendix by O. S. Kerr). J. Fluid Mech. 455, 1–19.Google Scholar
Chen, C. F. & Sandford, R. D. 1977 Stability of time-dependent double-diffusive convection in an inclined slot. J. Fluid Mech. 83, 83–95.Google Scholar
Fernando, H. J. S. 1987 The formation of a layered structure when a stable salinity gradient is heated from below. J. Fluid Mech. 182, 525–541.Google Scholar
Foster, T. D. 1965 Stability of a homogeneous fluid cooled uniformly from above. Phys. Fluids 8, 1249–1257.Google Scholar
Foster, T. D. 1968 Effect of boundary conditions on the onset of convection. Phys. Fluids 11, 1257–1262.Google Scholar
Holyer, J. Y. 1983 Double-diffusive interleaving due to horizontal gradients. J. Fluid Mech. 137, 347–362.Google Scholar
Huppert, H. & Linden, P. 1979 On heating a stable salinity gradient from below. J. Fluid Mech. 95, 431–464.Google Scholar
Kazmierczak, M. & Poulikakos, D. 1990 Transient double diffusion in a stably stratified fluid layer heated from below. J. Heat Fluid Flow 11, 30–39.Google Scholar
Kerpel, J., Tanny, J. & Tsinober, A. B. 1991 On a stable solute gradient heated from below with prescribed temperature. J. Fluid Mech. 223, 83–91.Google Scholar
Kerr, O. S. 1989 Heating a salinity gradient from a vertical sidewall: linear theory. J. Fluid Mech. 207, 323–352.Google Scholar
Kerr, O. S. 1990 Heating a salinity gradient from a vertical sidewall: nonlinear theory. J. Fluid Mech. 217, 529–546.Google Scholar
Kerr, O. S. 2000 The criteria for the onset of double-diffusive instabilities at a vertical boundary. Phys. Fluids 12, 3289–3292.Google Scholar
Kerr, O. S. & Gumm, Z. 2017 Thermal instability in a time-dependent base state due to sudden heating. J. Fluid Mech. 825, 1002–1034.Google Scholar
Kerr, O. S. & Tang, K. Y. 1999 Double-diffusive convection in a vertical slot. J. Fluid Mech. 392, 213–232.Google Scholar
Linden, P. F. & Weber, J. E. 1977 The formation of layers in a double-diffusive system with a sloping boundary. J. Fluid Mech. 81, 757–773.Google Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493–517.Google Scholar
Molemaker, J. & Dijkstra, H. A. 1997 The formation and evolution of a diffusive interface. J. Fluid Mech. 331, 199–229.Google Scholar
Proctor, M. R. E. 1981 Steady subcritical thermohaline convection. J. Fluid Mech. 105, 507–521.Google Scholar
Radko, T. 2013 Double-Diffusive Convection. Cambridge University Press.Google Scholar
Schladow, S. G., Thomas, E. & Koseff, J. R. 1992 The dynamics of intrusions into a thermohaline stratification. J. Fluid Mech. 236, 127–165.Google Scholar
Schmitt, R. W. 1994 Double diffusion in oceanography. Annu. Rev. Fluid Mech. 26, 255–285.Google Scholar
Stern, M. E. 1960 The ‘salt fountain’ and thermohaline convection. Tellus 12, 172–175.Google Scholar
Stommel, H., Arons, A. B. & Blanchard, D. 1956 An oceanographical curiosity: the perpetual salt fountain. Deep-Sea Res. 3, 152–153.Google Scholar
Tanny, J. & Tsinober, A. B. 1988 The dynamics and structure of double-diffusive layers in sidewall-heating experiments. J. Fluid Mech. 196, 135–156.Google Scholar
Terrones, G. 1993 Cross-diffusion effects on the stability criteria in a triply diffusive system. Phys. Fluids A 5, 2172–2182.Google Scholar
Thangam, S., Zebib, A. & Chen, C. F. 1981 Transition from shear to sideways diffusive instability in a vertical slot. J. Fluid Mech. 112, 151–160.Google Scholar
Turner, J. S. 1974 Double-diffusive phenomena. Annu. Rev. Fluid Mech. 6, 37–56.Google Scholar
Turner, J. S. & Stommel, H. 1964 A new case of convection in the presence of combined vertical salinity and temperature gradients. Proc. Natl Acad. Sci. USA 52, 49–53.Google Scholar
Young, Y. & Rosner, R. 1998 Linear and weakly nonlinear analysis of doubly-diffusive vertical slot convection. Phys. Rev. E 57, 5554–5563.Google Scholar