The growth rate of super-critical salt fingers

Abstract

An expression for the growth rate of super-critical salt fingers is derived from similarity solutions to the Boussinesq governing equations. The fastest growing fingers have an e-folding time of about one Brunt-Väisälä period in the ‘central waters’ of the world's oceans, with faster growth rates realized in water masses with larger salinity gradients, such as below the Mediterranean outflow. This rapid growth rate supports the notion that salt fingers must be nearly ubiquitous in the main thermo-halocline of the ocean. The flux ratio and wavenumber of the fastest growing fingers are also computed, with the explicit dependence on Prandtl number and diffusivity ratio retained. The wave number of the fastest growing fingers agrees with the oceanic observations of Magnell ( Journal of Physical Oceanography, 6, 511–523, 1976), and the flux ratio agrees well with the experimental data of Turner ( Deep-Sea Research, 14, 599–611, 1967) for heat and salt, as well as with the data of Stern and Turner ( Deep-Sea Research, 16, 497–511, 1969) and Lambert and Demenkow ( Journal of Fluid Mechanics, 54, 627–640, 1972) for sugar and salt. The good agreement in systems having two orders of magnitude difference in Prandtl number and diffusivity ratio suggests that time-dependent effects must play an important role in the dynamics of salt finger convection.