The salt finger amplitude in unbounded T-S gradient layers

Abstract

Finite amplitude numerical calculations are made for a completely unbounded salt e nger domain whose overall vertical ‘ ‘ property’ ’ gradients ( Tz andSz) are uniform and remain unaltered in time. For diffusivity ratio t 5 k S/k T 5 O(1), Prandtl number n /k T ae 1, and density ratio R 5 Tz/Sz . 1 this regime corresponds to a ‘ ‘ double gradient’ ’ sugar ( S) 2 salt (T) experiment. Two-dimensional pseudo-spectral calculations are made in the vicinity of the minimum critical condition for salt e nger instability, viz., small e ; (Rt ) 2 1 2 1 . 0; the allowed spectrumincludes the fastest growing wave of linear theory. When the vertical wavelength of the fundamental Fourier component is systematically increased the solution changes from a single steady vertical mode to a multi-modal statistically steady chaotic state. Each of the long vertical modes can be amplie ed by the (unchanging overall) gradient Sz, and can be stabilized by the induced vertical T, S gradients on the same scale as the modes; nonlinear triad interactions in the T 2 S equations can also lead to amplitude equilibration even though e , k T/n , and the Reynolds number are extremely small. When subharmonics of the horizonal wavelength of maximum growth are introduced into the numerical calculations the new wave amplie es (via Sz) and produces a quantitative change in the time average e uxes. Experimentally testable values of heat e ux and rms horizontal T-e uctuations are computed in the range2.8 . R . 1.6 for t 5 1 A3.Asymptotic similarity laws e ® 0 are also presented.