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

Kerr, O. ORCID: 0000-0003-2946-0695 (2018). Double-diffusive instabilities at a horizontal boundary after the sudden onset of heating. Journal of Fluid Mechanics, doi: 10.1017/jfm.2018.821

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 (2017) 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 efectiveness 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 t3=2, and for a constant heat flux it is proportional to t2, where t is the time since the onset of heating. However, the effective salt Rayleigh number also grows as t2. 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.

Publication Type: Article
Additional Information: This article has been published in a revised form in Journal of Fluid Mechanics https://doi.org/10.1017/jfm.2018.821. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © Cambridge University Press.
Subjects: Q Science > QA Mathematics
Departments: School of Mathematics, Computer Science & Engineering > Mathematics
URI: http://openaccess.city.ac.uk/id/eprint/20710
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