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The growth and structure of double-diffusive cells adjacent to a cooled sidewall in a salt-stratified environment

  • LIORA MALKI-EPSHTEIN (a1), OWEN M. PHILLIPS (a2) and HERBERT E. HUPPERT (a1)

Abstract

Observations and measurements are reported on the patterns and rates of growth in time of the double-diffusive cells that form adjacent to a cooled sidewall in a saltstratified environment. Fluid near the wall is cooled and sinks a distance $h$ where its density, increased by cooling, matches that of the salt-stratified ambient. The fluid separates from the wall, moving outwards as a cool, fresher layer beneath a warmer, more saline region. This leads to growing double-diffusive cells that advance outward at a rate, found by dimensional reasoning, to initially be proportional to $N_{0}h,$ where $N_{0}$ is the initial buoyancy frequency in the ambient and $h$ is the intrusion's vertical thickness. Near the wall at the top of each cell, the sinking colder fluid is continually replaced by selective withdrawal from the ambient ‘far field’. The fluid being withdrawn from the ambient is always the least dense in the cell, and as the experiment proceeds, the straining of the fluid in the ambient region reduces the stratification. The vertical density gradient inside the cell relaxes by continuous hydrostatic adjustment (CHA) to match the ambient and the speed of advance reduces. Measurements of the rate of advance of the cell nose were made in tanks of different lengths $L$ with a range of initial salinity gradients and temperature differences. A simple two-dimensional model is developed to describe the rate of extension of the cells and the internal density gradient as functions of time in which the tank length appears as an important variable. This effect does not seem to have been recognized previously. The rates of evolution in each run involve the time scale $\tau \,{=}\, L /( {C_H hN_0 })$, where $C_H \,{\approx}\, 10^{ - 2}$ is a heat transfer coefficient. The mean length of the cells $\skew2\bar {l}(t)$and the internal buoyancy frequency as functions of time are given by \[ \skew2\bar {l}(t) / L = t/\tau - ( t/2\tau)^2,\quad N = N_0 (1 - t / 2\tau ). \] Inversion of the first of these expressions results in $t/\tau \,{=}\, 2\,{-}\, 2\{ {1 - (\skew2\bar {l}(t) / L)}\}^{1 / 2}$ from which a time scale $\tau ^{ - 1}$ can be estimated. The measurements from individual runs when plotted in this way generally produce accurate straight lines as the model predicts, from which $C_H $ is found. This should be approximately the same for each run; the mean over all runs was found to be $9.3\,{\times}\, 10^{ - 3}$ with standard deviation 2.4$\,{\times}\, 10^{ - 3}$. The velocity scale of the intrusions at the beginning of an experiment is of order 10$^{ - 2}$ cm s$^{-1}$, for typical parameters of water at temperature 20 $^\circ$C, cooled wall temperature of 0 $^\circ$C and mean salinity of 5%.

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The growth and structure of double-diffusive cells adjacent to a cooled sidewall in a salt-stratified environment

  • LIORA MALKI-EPSHTEIN (a1), OWEN M. PHILLIPS (a2) and HERBERT E. HUPPERT (a1)

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